Material property prediction

Abstract

Material property prediction method comprise modelling a periodic structure as a quotient complex (QC). The QC models pairwise and many-body interactions using simplices of plurality of dimensions, and also models periodic material properties by applying a quotient operation over the simplices. A feature vector is then generated for each simplex in the QC, and is updated for each simplex in the QC based on feature vectors of one or both of neighbouring simplexes in the QC and cofaces of the respective simplex, using a message passing scheme. The QC can then be processed with updated feature vectors using a simplex transformer to predict material properties of the periodic structure.

Description

[0001] MATERIAL PROPERTY PREDICTION

[0002] Technical Field

[0003] The present invention relates to the field of computational material science and machine learning technology. More specifically, the invention relates to systems and methods for predicting material properties, such as the properties of crystal structures, using simplex-based transformers and graph-based models.

[0004] Background

[0005] The reference in this specification to any prior publication (or information derived from it), or to any matter which is known, is not, and should not be taken as an acknowledgment or admission or any form of suggestion that that prior publication (or information derived from it) or known matter forms part of the common general knowledge in the field of endeavour to which this specification relates.

[0006] Recent advancements in computational material science have significantly accelerated the discovery and optimization of novel materials. This field leverages mathematical modelling, physics-based simulations, and machine learning techniques to predict material properties without relying solely on timeconsuming experimental methods. Applications range from energy storage systems, such as batteries and fuel cells, to advanced structural materials used in aerospace and electronics. In particular, the integration of machine learning with material science has enabled researchers to analyse complex datasets and identify patterns that were previously difficult to discern. As computational power continues to grow, there is an increasing focus on developing models capable of interpreting intricate material structures, including their atomic arrangements and periodic properties, to achieve higher accuracy and efficiency. Despite these advancements, existing approaches face several limitations when applied to the prediction of material properties. Traditional methods often rely on simplified representations of crystal structures, which fail to adequately capture the complexity of interactions within a material. For instance, many models emphasize pairwise interactions while neglecting higher-dimensional (also referred to as "higher-order") relationships between atoms or bonds, leading to incomplete descriptions of material behaviour. Furthermore, some techniques struggle to incorporate periodicity, which is essential for accurately modelling crystalline materials. These constraints hinder the ability to predict properties with high precision, particularly for systems involving multi-body interactions or complex geometric configurations. Additionally, conventional algorithms may require extensive computational resources, limiting their scalability and practical applicability in large-scale material screening processes.

[0007] It would be desirable to provide a framework for predicting properties of materials that takes into account structural material features and periodic material features and boundary conditions.

[0008] Summary

[0009] To address the abovementioned challenges, the present framework accounts for both local and global interactions within a structure, particularly a lattice or crystalline structure, while efficiently handling periodic boundary conditions. This framework enables information to be updated across different levels of structural granularity, enabling the integration of diverse types of data and a broad range of interactions between atoms and bonds in the structure.

[0010] The framework described herein predicts material properties with high accuracy by modelling crystal structures as a quotient complex (QC). The QC captures both pairwise and many-body interactions using simplices of multiple dimensions. Moreover, that framework efficiently updates feature vectors of simplices of the QC through a message-passing scheme, and processes those feature vectors using a simplex-based transformer. This enables precise prediction of periodic material properties while accounting for local and global structural interactions.

[0011] The invention also improves computational efficiency and scalability for large-scale material screening applications. This is achieved by applying a hierarchical message-passing mechanism across different simplex dimensions. The integration of attention mechanisms ensures precise information exchange between neighbouring simplices and cofaces, enabling comprehensive analysis of geometric and topological relationships within crystal structures.

[0012] Disclosed is a material property prediction method comprising:

[0013] modelling a periodic structure as a quotient complex (QC), the quotient complex modelling:

[0014] pairwise and many-body interactions using simplices of plurality of dimensions; and

[0015] periodic material properties by applying a quotient operation over the simplices;

[0016] generating a feature vector for each simplex in the QC;

[0017] updating the feature vector for each simplex in the QC based on feature vectors of one or both of neighbouring simplexes in the QC and cofaces of the respective simplex, using a message passing scheme; and

[0018] processing the QC with updated feature vectors using a simplex transformer to predict one or more material properties of the periodic structure.

[0019] Also disclosed is a system for predicting material properties of crystal structure materials, comprising:

[0020] memory;

[0021] at least one processor;

[0022] a quotient complex modeller; and

[0023] a transformer module, wherein the memory stores instructions that, when executed by the at least one processor, cause the:

[0024] quotient complex modeller to receive input comprising a periodic structure, model pairwise and many-body interactions using simplices of plurality of dimensions, model periodic material properties by applying a quotient operation over the simplices, and generate a quotient complex (QC) from the modelled interactions and periodic material properties;

[0025] transformer module to:

[0026] generate a feature vector for each simplex in the QC; update the feature vector for each simplex in the QC based on feature vectors of one or both of neighbouring said simplexes in the QC and cofaces of the respective simplex, using a message passing scheme; and

[0027] process the QC with updated feature vectors using a simplex transformer to predict one or more material properties of the periodic structure.

[0028] Brief description of the drawings

[0029] Embodiments of the present invention will now be described, by way of nonlimiting example, with reference to the drawings in which:

[0030] FIG. 1 illustrates a method for predicting material properties using simplexbased transformers according to an embodiment of the present disclosure.

[0031] FIG. 2 schematically illustrates a crystal structure (a), typical graph (b) and simplicial complex (c) representation of the structure (a), a quotient graph (d) and QC (e) in accordance with an embodiment of the present disclosure, and QC representations of 0-, 1- and 2-dimensional simplices (f).

[0032] FIG. 3 schematically shows the QC Transformer (QCformer) architecture, including the overall framework of the QCformer (a), the feature embedding block (b) of the framework (a), and the Simplex Transformer (Sformer) block (c) in framework (a).

[0033] FIG. 4 illustrates simplicial complexes and their corresponding quotient complexes. The quotient operation merges equivalent vertices in the simplicial complexes, resulting in the derived quotient complexes.

[0034] FIG. 5 depicts the construction of the simplicial complex eK, which is a homotopic equivalent of the quotient complex K.

[0035] FIG 6 illustrates topological spaces and continuous maps, including homotopy equivalences.

[0036] FIG 7 illustrates topological spaces and a constant map.

[0037] FIG 8 illustrates crystal quotient complex construction with three nearest neighbours.

[0038] FIG 9 illustrates quotient complex construction on ZnS.

[0039] FIG 10 shows bandgap predictions for unlabelled 2D perovskite materials using the framework of FIG 3.

[0040] FIG. 11 illustrates a system for performing material property predictions in accordance with present teachings.

[0041] Detailed description

[0042] The present modelling and prediction framework predicts material properties with high accuracy by modelling periodic structures and periodic-like structures, such as crystal structures, as a QC. For illustration purposes, the following description will focus on modelling crystal structures, though the same teachings apply to the aforesaid periodic-like, or near-periodic, structures such as polymers. For ease of understanding, the term "periodic structure" will be taken to include crystal structures, polymers and other near-periodic or periodic-like structures - i.e., structures with particular repeating components, where other components of the same structures are non-periodic (e.g., polymers with nonperiodic active groups). The QC captures both pairwise and many-body interactions using simplices of multiple dimensions, and incorporates periodic structural information through a quotient operation over the simplices. Simplex feature vectors of QC simplices are updated through a message-passing scheme, and processed using a simplex-based transformer for precise prediction of periodic material properties while accounting for local and global structural interactions.

[0043] Referring to FIG. 1, the prediction method (100) is depicted, for modelling a structure as a QC (102) and using the QC to predict material properties. Some components of the prediction method (100) according to this embodiment may be modified or added to the components shown in FIG. 1, and it is to be understood that various modifications and changes can be made without departing from the technical spirit of the present invention.

[0044] The method (100) involves with modelling interactions between bodies (104), which serves as the foundation for understanding the behaviour of the crystal structure. The crystal structure is modelled (102) by representing the crystal structure as a simplicial complex, where vertices, edges, and higherdimensional simplices are used to model interactions. This step allows the representation of pairwise and many-body interactions using simplices of plurality of dimensions, enabling the capture of complex structural relationships and topology within the material. Each simplicial complex (simplex - pl. simplices) is a structured set of points, line segments, triangles, and their n-dimensional or higher-dimensional counterparts. A resulting simplicial complex includes all faces and intersections of the elements in the set. Modelling simplices will be understood by the skilled person after a review of graph theory shape and structure modelling.

[0045] Modelling periodic material properties (106) involves applying a quotient operation over the simplices to represent periodicity in the material. This step characterizes the periodic attributes of a material, capturing periodic equivalence relations defined over the crystal structure, which is critical for predicting material properties. In particular, equivalent simplices are merged in the resulting QC. This merging process simplifies the representation while preserving the essential structural and periodic properties of the material. The resulting quotient complex (QC) provides a compact yet comprehensive model of the crystal structure, which can be used for further analysis and prediction.

[0046] After modelling periodic properties using a quotient operation, and thereby forming the components (simplices) of the QC (nodes, edges, faces and higherdimensional components), feature vectors are generated (108) for each simplex in the QC - a simplex "in the QC" represents all simplices equivalent to it in the original simplicial graph. The feature vectors encapsulate the characteristics of the modelled interactions and material properties. Since the QC merges repeating simplices, each simplex in the QC is unique relative to all other simplices in the QC. Thus, each simplex in the QC is assigned a feature vector that encapsulates its unique characteristics.

[0047] During transformer training, feature vectors may include the quantities sought to be predicted by future applications of the trained system. Feature vectors can also include higher level embeddings between data, such as the bandgap between the conduction band and valence band of a crystalline structure, rather than the conduction and valence bands themselves, temperature-dependent characteristics and others.

[0048] The feature vectors are generated using one-hot encoding for vertices and radial basis functions (RBFs) for other simplices, ensuring a comprehensive representation of the crystal structure. Other operations may be used in place of one-hot encoding and RBFs, such as label encoding, multiquadric, inverse quadratic, Matern kernels and other Gaussian processes, Fourier functions and others. The encoding methods are selected to ensure representation of topological and geometric properties of the crystal structure.

[0049] Feature vectors are then updated (110) to refine the feature vectors (initial representations) by incorporating additional data or insights from neighbouring simplices and cofaces. The updating process uses a message passing scheme, where messages flow from higher-dimensional simplices to lower-dimensional simplices, ensuring that the feature vectors remain optimized for accuracy. In some embodiments, the order of insertion may change such that the message passing scheme uses messages flowing from lower-dimensional simplices to higher-dimensional simplices. In some embodiments, attention coefficient vectors are computed between each simplex and one or more, and preferable all, other simplices in the QC. The attention mechanism more heavily weights relevant interactions between simplices, over interactions that are less influential in the overall properties of the crystal structure. The messages passed in the message passing scheme are then computed based on the attention coefficients, and used to update the feature vectors. The updated feature vectors provide a more precise characterization of the properties of the crystal structure, including dynamic interactions within the crystal structure.

[0050] Processing the QC, with now updated feature vectors, (112) facilitates prediction of the material properties of the crystal structure. This step utilizes a simplexbased transformer architecture to analyze the updated feature vectors and predict material properties. The transformer processes the feature vectors through a series of Node Transformer and Edge-wise Transformer layers, capturing both local and global structural information.

[0051] During the training stage, gradients derived from a loss function over the output of a prediction layer (typically the output layer of a Multi-Layer Perceptron (MLP)), are back-propagated through the network - simplex transformers, MLP and / or other components - to refine prediction accuracy. We use the AdamW optimizer with a weight decay of le-5 and a one-cycle learning rate scheduler, while the mean squared error is employed as the loss function to train the model. Other loss functions, such as least squares, may be used in place of gradients. During the prediction stage, the trained simplex-based transformer architecture, herein referred to as the QCformer, predicts one or more material properties of the crystal structure.

[0052] The transformer processes the updated feature vectors through a sequence of Node and Edge-wise Transformer layers in a hierarchical manner discussed with reference to FIG. 3. This hierarchical processing ensures that both local and global structural information is effectively captured and utilized for property prediction, such as the prediction of valence, cohesion and bandgap values.

[0053] The method (100) is designed to achieve high accuracy in predicting material properties, such as bandgap values for unlabelled 2D perovskite materials, as illustrated in FIG. 10.

[0054] FIG. 2 illustrates a method for modelling the crystal structure as a QC and constructing a crystal quotient graph. As mentioned above, the quotient complex represents pairwise and many-body interactions using simplices of various dimensions, and periodic material properties are modelled by applying a quotient operation over the simplices. The crystal structure (a) can be visualised as an infinite graph in which vertices are atoms and edges are bonds or "interactions" between atoms. The compound lattice shown in (a) is CaT103, though the present teachings can be applied to materials more generally. For example, the present teachings can be applied to near-periodic, or periodic-like, structures such as polymers.

[0055] In particular, a crystal structure C can be represented by a unit cell with periodic patterns. A unit cell is the minimal repeatable structure of a given crystal and can be described by two matrices A and P. Focussing on 3D crystal structures, matrix P = [pltp2, ■■■,pn\T elRnx3 isthe coordinate matrix where pte R3is the 3D Cartesian coordinate of the ithatom in the unit cell. Matrix A = [alta2,...,an] e Znis the atom type matrix where atis the atomic type of the ithatom. For the periodic patterns, a lattice matrix L = [, l2, 13V e R3x3is used to describe how the unit cell repeats itself in three directions Zi2 3to form the whole crystal structure. Given the crystal representation C = (A, P, L), the infinite crystal structure can be represented as

[0056] C = {(jpi, &i) | &i = a^pt = Pt + k±l±+ k2l2+ fc3Z3,fc1, k2,k3e Z, 1 < i < n (1)

[0057] where ptcorresponds to all periodic translations of pt, and all such translations retain the atom type at. The goal of the crystal property prediction task is to learn a function f-. C -> IR that maps a crystal structure to its property value. The essential assumption for this function is the molecular structure-functional relationship. That is, if the molecular structures are well understood, their various properties and functions will be able to be explained.

[0058] A finite crystal graph (b) can be extracted from the structure (a), where nodes of the graph (b) thus represent Ca (Calcium), Ti (Titanium), and 0 (Oxygen) atoms. The edges represent bonds between these atoms. This graph simplifies the connectivity between atoms while retaining essential structural information, by merging or gluing equivalent atoms together. The arrangement of nodes and edges may vary depending on the specific crystal structure being analysed.

[0059] A simplicial complex (c) can also be extracted from this infinite crystal graph by considering only the atoms within a defined supercell (composed of multiple unit cells) or the k-nearest neighbours of atoms within the unit cell. The number k may be determined or optimised empirically and, for present purposes, is assumed to be 12. The number k may also be set based on computational cost - e.g., k may be 10, 12 or even 20. For illustration, a supercell-based graph (b) is composed of 2x2x2 unit cells. By gluing or merging equivalent vertices together, a quotient graph (d) is constructed. The quotient graph (d) may be visualized as a reduced version of the original graph (b). It retains key connectivity information while eliminating redundant details. In contrast, a simplicial complex (c) can be constructed based on the points in the supercell.

[0060] As mentioned above, a generalised concept of a simplicial complex K over a finite vertex set V is a collection of non-empty finite subsets of V such that if a e K and r a, then r e K. Any element a e K with n + 1 vertices is termed an n-simplex, where n is also known as the dimension of a. The dimension of an abstract simplicial complex is defined as the highest dimension among its simplices. An abstract simplicial complex serves as the basis for determining real simplices embedded in Euclidean space, with a particular emphasis on the face relations between simplices. Specifically, any subset T of a is referred to as a face of a, and a is denoted as a coface of r.

[0061] Geometrically, O-simplices correspond to vertices and 1-simplices to edges, with shaded or filled-in areas forming 2-simplices corresponding to triangles, 3- simplices to tetrahedra, and so forth for higher-dimensional simplices. The embedding of an abstract simplicial complex into Euclidean space is referred to as the geometric realization. X denotes the embedded space and can simply be referred to as a simplicial complex. In particular, since V is a finite set, the embedded X is a compact space.

[0062] For ordered simplicial complexes, a set K is defined over a finite vertex set V, forming an abstract simplicial complex with orderings on its simplices. Mathematically, each n-simplex in K is represented as an ordered sequence a = (0,v1,...,vn) of n + 1 vertices. Furthermore, every simplex represented as a subsequence of <r is also required to be an element in K. In particular, for any vn) and i e {0,l,...,n}, the ordered subset a‘: =

[0063]

[0064] ,,t,,n,., Vj-i, vi+1,..., vn), obtained by removing the ithvertex from a, is called the ithface of a. Similar to an abstract simplicial complex, <7 is a coface of the simplices <. Two n-simplices o and T are called upper adjacent if they share a common coface, denoted by o — T.

[0065] An undirected graph can thus be generalized to a simplicial complex, and a directed graph can be generalized to an ordered simplicial complex. Specifically, both of them are abstract and ordered simplicial complexes of dimension at most 1. Beyond the interactions encoded by edges, higher-dimensional simplices are capable of capturing more complex and intricate interactions involving more than two nodes or vertices.

[0066] Directed graphs also contain clique complexes. Given a directed graph G = (V, E) with vertex set V and arrow set E. A clique o of G is a subsequence a = (io,vi, vi2, -'vin) °vsuch that vi vike E for all j < k. All the cliques of G form an ordered simplicial complex, which is called the clique complex of the directed graph G. For example, if the directed graph G = (V, E) consists of three vertices 1,2 and 3 and three arrows (1,2), (1,3), (2,3), then the associated clique complex is

[0067] {(1),(2),(3), (1,2), (1,3), (2,3), (1,2,3)}, which contains an ordered 2-simplex (1,2,3). However, if the arrows set is defined by {(1,2), (2,3), (3,1)}, then the induced clique complex contains no ordered 2-simplices.

[0068] Using conceptually the same equivalence relations as were used to convert the finite crystal graph (b) to the quotient graph (d), a quotient complex (e) can be constructed from the simplicial complex (c). This quotient complex (e) encompasses multiple edges to account for pairwise interactions between distinct vertices, self-loops representing self-interactions of individual vertices, and triangles that indicate higher-dimensional interactions involving three distinct vertices. Thus, the quotient complex can be seen as a generalization of the quotient graph. The same process can be used to represent multiscale interactions within or between atoms of unit cells can be represented by a series of nested QCs through a filtration process. To that end, the method may comprise setting a cutoff distance (the number of atoms between any two atoms, including the two atoms themselves) to define edges between any two nodes. The cut-off distance can then be systematically increased to define a series of structures. For instance, for the same point cloud data (from materials), a cutoff distance of 4\AA (a series of four atoms) can be used to generate the first QC, then a cut-off distance of 5\AA (a series of five atoms) can be used to generate a second QC, to thereby generate nested QCs. The method may therefore include changing the cut-off distance (e.g., incrementing or decrementing by one or more atoms) to generate a series of nested QCs. Thus, where a subsequent action is performed on a QC - e.g., modelling pairwise and many-body interactions using simplices of plurality of dimensions, modelling periodic material properties by applying a quotient operation over the simplices, and generating a feature vector for each simplex in the QC - these steps can be used for each nested QC. The subsequent training steps can be similarly applied to each nested QC - i.e., the term "QC" can be substituted for "each nested QC" or similar, as permitted by context.

[0069] The quotient operation used to generate the quotient graph (d) from the finite crystal graph (b) and the quotient complex (e) from the simplicial complex (c), thus generates new topological spaces from a given topological space. For a topological space x, an equivalence relation ~ on x can be defined to obtain the set of equivalence classes X / ~= {[x]:x e X} along with the canonical quotient map q X -> X / ~, where x [x]. The quotient topology on X / ~ is the finest topology that makes the map q continuous. Focussing on simplicial complexes K embedded in the D-dimensional Euclidean space IRD, equivalence relations ~vcan be defined on the vertex set V to examine the induced quotient complex % / ~7.

[0070] Figure 4 shows the conceptual conversion of simplicial complexes to quotient complexes. In these examples, vertices of the same types are equivalent. The vertices represent O-dimensional simplices. The first row of Figure 4 shows the simplicial complexes and the second row shows the corresponding quotient complexes, in which equivalent vertices (O-dimensional simplices) are merged. For instance, in (a), the simplicial complex consists of 5 equivalent vertices and 4 edges. Since all vertices are equivalent, the induced quotient complex has only one vertex and four self-loops. The simplicial complex in (b) includes 5 vertices of three types (a vertex is of the same "type" as another vertex if they are equivalent), 5 edges, and 1 triangle. The 5 vertices are classified into three equivalence classes, inducing the quotient complex shown in the second row. The same process applies for (c), with the induced quotient complex including two "torsion" triangles, sharing the same vertices.

[0071] Exploring equivalence, let x denote a topological space and A a subset of x. For a continuous map f: A -> Y, one can define the adjunction space denoted as X ufY. This space is constructed as the quotient space X u Y / ~, where X u / denotes the disjoint union space of X and Y, and ~ represents the least equivalence relation generated by a~f(a) for all a e A Q X. The quotient operation enables constructing of cell complexes.

[0072] As a specific instance of a quotient space, certain special adjunction spaces X u;Y can be realized through a more straightforward equivalence relation applied to the space X. Given a crystal simplicial complex representation K with vertex set A, let z be the set of equivalent classes of A under periodic equivalent relation —period ■ There is a natural projection f-. A ^ Z. Then the adjunction space X u;Y is exactly the quotient complex derived from K by identifying equivalent atoms up to —periQd ■ Denoting X uzY by %, this insight facilitates a convenient technique for identifying and gluing or merging specific points within X and enables exploration of quotient complexes. For example, the following proposition offers an alternative way to represent the quotient space X / ~A. In particular, let x and Y be topological spaces, A a subspace of x, and f -. A -> Y a continuous map. Then there is a continuous map <p-. X / ~A- XvfY that maps every equivalence class [x] in X / ~Ato the equivalence class [x] in X uzY. Furthermore, if f is onto and Y is discrete, then <p is a homeomorphism. Specifically, let K be the realization of K over a finite vertex set V, ~van equivalence relation on v, and z a discrete topological space. Suppose there is a surjective function f: V -> Z such that (v) = f(w) if v ~vw. Notably, Z is finite since there are only finitely many equivalence classes in V. Then the induced quotient complex X is homeomorphic to the adjunction space

[0073]

[0074] Z. This identification proves useful for the subsequent exploration of the topological structure of the quotient complex.

[0075] Further in this regard, for the finite simplicial complex previously elucidated, a set of vertices, V, was delineated within the complex K. For geometric realizations X, the quotient spaces X were created by merging equivalent points in V, separately. Similarly, a method is available for constructing a simplicial complex, referred to as X, which is homotopy equivalent to X. This method is depicted in FIG. 5, which depicts the construction of the simplicial complex X, which is homotopy equivalent to the quotient complex X. In this example, X denotes a simplicial complex with 6 vertices., 7 edges, and 1 triangle. The vertex set V is divided into three groups: VlfV2, and V3. By adding star-shaped sets

[0076]

[0077] and S3to the simplicial complex x, the simplicial complex X is obtained, which is homotopy equivalent to the quotient complex X.

[0078] To better elucidate the homotopy equivalence relation between x and the quotient complex x, suppose FIG 6, being a diagram of topological spaces and continuous maps, commutes, where <px,<f>A,<pyare homotopy equivalences, and the inclusions i,i' are closed cofibrations. Then the map p XfY X'

[0079]

[0080] Y' induced by the maps <px, <pA,<pyis a homotopy equivalence.

[0081] The mathematical formulation of the space x is established as follows. Let vlr...,vkbe disjoint subsets of the vertex set V of x. Let z1,z2,...,zfcin RD I' such that z7+1e RD+ / +1\ RD+;, where RD+;is identified as a subspace of RD+ +1via the canonical embedding map ( x1,...,xD+j)

[0082]

[0083] ...,xD+;,o). Setting V = |_|J=1V}and z-. = {z^z-,...,zk}, which is a discrete space, the desired space that emerges points in v, to a single point can be identified as the adjunction space x Jtz = X uf{z^z^...,zk, where f-. V ^ Z is the local constant function with (V7) = {z. Next, for each j e {1,2,...,k} a star-shaped set Sj is defined as the union of all line segments that link z;to points in Vj. That is,

[0084]

[0085] In particular, Stns;= 0 for i j e {1,2,...,k by the assumption of z1,z2,...,zfc. Set s = U;=iSj and define g-.v - s such that g\vequals the inclusion map Vj <^s. Finally, set K = uS = u((S. Then FIG. 7 commutes, where r S -> Z is defined such that r|sequals the constant map Sj -> {z7} <-> Z. Since the continuous maps r|scan be shrunk synchronously, r is a homotopy equivalence. Because any inclusion map of a subcomplex of a simplicial complex is a closed cofibration, i is a closed cofibration.

[0086] It therefore follows that letting K, v = |_|=1Vj,x,x, and x be defined as above, then X and X are homotopy equivalent. In particular, the homology of X and X are isomorphic. In other words, there is a homotopy equivalence f from x u5s = x u s = x to x uzz = K, which is induced by the homotopy equivalence r S -> Z.

[0087] This is visually demonstrated in FIG. 5. Leveraging the simplicial complex X and the vertex set V = Lly=1Vj, the simplicial complex X is constructed as the union XuS, with s = s1us2us3denoting the union of star-shaped sets as illustrated in FIG. 5. As determined above, both the simplicial complex X and the quotient space X share the same homotopy type. Notably, X and X exhibit identical homology groups. As a result, there is a method for representing X by X up to homotopy equivalence, and the inclusion relationship x c% is established. This representation offers a combinatorial perspective to the study of persistent homology within quotient complexes, thereby facilitating the analysis and proof of persistent barcode information.

[0088] As a result of the construction presented previously, the space K assumes the form of a simplicial complex within RD+fe, with X representing a subcomplex of X. Consequently, the inclusion map X <-»X induces a homomorphism denoted as e.: H(X) -> H(X). A theorem can then be stated:

[0089] d X be defined as above. Then:

[0090] onto;

[0091] one-to-one;

[0092]

[0093] an isomorphism for q > 1.

[0094] (a) and (b) elucidate the relations of the homology groups between the quotient complex X - X and the original simplicial complex X in dimensions 0 and 1. In particular, (a) and (b) state that the quotient complex x retains the topological information that is invariant under periodic equivalence in the crystal simplicial complex X, while eliminating redundant topological information, (c) asserts that the quotient complex x encapsulates the same homological information as X. In particular, the quotient operation acts specifically on vertices. The reason is that the gluing or merging operation is only applied to O-dimensional simplices. Thus, (a) to (c) demonstrate that the quotient complex captures periodic patterns inherent in crystal structures.

[0095] With further reference to FIG. 2, the simplicial complex (c) connects to the quotient complex (e) through a quotient operation. Once suitable equivalence relation(s) are defined between simplices, a QC (e) can be constructed from a simplicial complex (c), by gluing or merging the vertices / edges / higher-dimensional simplices together based on periodic equivalence relations. Equivalence of atoms and simplices is determined by reference to the nature of bonds with neighbouring atoms and simplices, next nearest neighbours (i.e., neighbours of neighbours and so on) as needs and computational resources permit. The resulting quotient complex (e) simplifies the representation while preserving essential properties.

[0096] More particularly, the process of constructing a quotient complex representation of an infinite crystal structure involves building a directed graph from the crystal structure, adding triangles to the directed graph to form a simplicial complex, and applying a quotient operation to the simplicial complex to obtain the final quotient complex.

[0097] In an illustrative embodiment, k -nearest neighbour graph construction generates the graph. Formally, let the atoms of a unit cell form the vertex set 7, and denote the nearest k neighbors of a vertex v by Nfe(v). The directed edge set E is defined

[0098]

[0099] e v}. The pair ( V, E ) then forms a directed graph.

[0100] Clique complex of the directed graph are then used to incorporate triangles. Specifically, for a triangle

[0101]

[0102] with an orientation (an order) where £', a total order is assigned to these three edges as

[0103]

[0104] . Finally, forthe quotient operation, we define two atoms a and b as equivalent if there exist k,k2,k3e Z such that a = b + k1l1+ k2l2+ k3l3rwhere lltl2,l3are the lattice vectors. The quotient complex C is derived by gluing all equivalent atoms together.

[0105] FIG. 8 gives an illustration of crystal quotient complex construction for / c = 3. First, the atoms within the unit cell (800a)-(800c) are selected as nodes, with directed edges added from the three nearest neighbours to each target node (800a)-(800c). Next, triangles (802) are formed based on edge directions (where edge directions form a loop, it is not considered a triangle), constructing the simplicial complex. Finally, nodes of the same type are merged to form the quotient complex (804).

[0106] The structure (a) of FIG. 2 is thus modelled as a QC (e). Step (108) then involves generating feature vectors. Feature vectors are generated using the QC Transformer (QCformer) model. Once feature vectors are generated, simplex features are updated by aggregating information from their neighbours and cofaces. There are various methods to define the neighbours of a simplex, such as upper adjacent, lower adjacent, parallel adjacent, and so on. In the QCformer model, the neighbours of each vertex a0= (v) are defined as N(a0~) = . Similarly, the neighbors of each edge

[0107]

[0108] = (y^Vj) are defined as

[0109]

[0110]

[0111] indicates that

[0112]

[0113] are upper adjacent.

[0114] In one embodiment, a k-nearest-neighbour-based crystal graph is constructed, e.g., where k = 12, and its clique complex is used to derive the quotient complex through the quotient operation on equivalent atoms. The crystal graph can be processed using a crystal graph convolutional neural network, in a known manner, to predict material properties.

[0115] A detailed example of the quotient complex construction for ZnS material is provided in FIG. 9. On the left-hand side, the unit cell structure is visualized using the VESTA program, though other programs may be used. On the righthand side, a graph is constructed by selecting the 12-nearest neighbours of each atom in the unit cell. As both the graph and the simplicial complex are three-dimensional objects, the two-dimensional projections shown have overlapping vertices and edges. The quotient graph and quotient complex are constructed by gluing equivariant vertices together.

[0116] Using a crystal graph convolutional neural network (CGCNN), features can be assigned to simplices. For illustration purposes, the process of assigning feature vectors to simplices will employ a 2-dimensional strategy, thus incorporating features for vertices (O-dimensional), edges (1-dimensional) and triangles (2-dimensional). For each vertex, the CGCNN feature is considered, which is a multidimensional vector - presently, a 92-dimensional vector. For each edge, its Euclidean length d is first mapped to d' where, for present purposes, d’ = -0.75 / d and then processed through the radial basis function (RBF). The present RBF is defined as rbf(d') = exp ■ The RBF expands d' into a higher

[0117]

[0118] dimensional vector, presently a 192-dimensional vector ( 64 x 3), where c ranges between a lower bound, presently -4.0 and an upper bound, presently 0.0, with a predetermined step size, e.g., 1 / 64. a takes values 0.01,0.1 and 1, or other values as desired. This RBF vector is then concatenated with the multidimensional CGCNN feature vectors of the two vertices connected by the edge, resulting in a multidimensional edge embedding, presently comprises 376 features ( 192 + 92 x 2 ). For triangles, combinations of edge lengths d1,d2,d3, defining the triangle, are used to generate additional features. These additional features may include d1■ d2,d±■ d3,d2■ d3, and / or the squares d1■ d^d- ■ d2,d3■ d3. These values, of which there are 9 in the present embodiment, are passed through the RBF, defined as rbf(a) = exp

[0119]

[0120] , to expand into a multidimensional vector, where c ranges from 0.0 to 5.0 with a step size of 1 / 8, and o takes values 0.01,0.1 and 1. The multidimensional vector has 216 dimensions ( 8 x 3 x 9 ) and serves as the triangle embedding.

[0121] With feature vectors now established per step (108), they can be updated per step (110). This process is illustrated in FIG. 3, where the QC is constructed (300), features are embedded based initially on a crystal graph visualization (302) and are then updated. The feature vector for each simplex in the QC are updated based on feature vectors of one or both of neighbouring simplexes in the QC and cofaces of the respective simplex, using a message passing scheme. The updating process is managed using Simplex Transformer (Sformer) modules. The Sformer module updates simplex representations based on information from neighbouring simplices and cofaces - the update process may aggregate features of those neighbouring simplices and cofaces. In particular, as shown in FIG. 3, (c), each n -simplex anis updated by aggregating information from its neighbors (n-simplices) rnand cofaces ((n + l)-simplices) (σn, τn)sand higher dimensional simplices are updated to be inputs for the updating of lower dimensional simplices.

[0122] To determine which features to update, an attention mechanism is employed. Any appropriate attention mechanism may be used. For present purposes, a standard scaled dot-product attention mechanism builds the aggregation function for aggregating feature vectors from neighbouring simplices and cofaces - in particular, the scaled dot-product attention mechanism computes attention coefficients between an n-simplex and its neighbours. The messagepassing scheme used to update the feature vectors requires computing an attention coefficient vector between each simplex and all other simplices in the QC, message computation based on that attention coefficient vector, and then simplex feature updating using the computed message.

[0123] Denoting the constructed crystal quotient complex, being the quotient complex of the crystal structure in question, as C, for any n-simplex ane C, assume that Tnis a neighbor of an. Two n-simplices are considered neighbours if they both serve as faces of a shared ( n+ 1 )-simplex, referred to as a coface. Unlike simplicial complex attention models, presently, two n -simplices that are neighbours can share multiple common cofaces. For instance, two vertices may share several edges connecting them. The sthcommon coface of anand rnis denoted by (σn, τn)s. Let

[0124]

[0125] be the feature vector for the n-simplex an. Its query q, key k and value v can be expressed as:

[0126]

[0127] where Qn, Kn, and Vnare the query, key, and value matrices for n-simplices, respectively. Matrices Kn+1and Vn+1are the key and value matrices for ( n + 1 )-simplices. Next, the message from

[0128]

[0129] to anis computed through (σn, τn)s:

[0130]

[0131] where d(**q**σ) is the dimension of **q**σ, aa-n: Tti)sis the attention coefficient vector of Tnto anthrough their sthcommon coface (an, Tn)s. The symbol ° represents the Hadamard product, and [ x,y ] represents the concatenation of x and y. Sig denotes the Sigmoid activation function, BN represents the Batch Normalization operation, and MLP is the Multi-Layer Perceptron used to compute the updated message. Here, the Matformer model is followed for ease of computation, and Sigmoid and Batch Normalization are used instead of the Softmax operation.

[0132] Finally, the updated feature vector of σnin ( l + 1 )thlayer is computed:

[0133]

[0134] where N(σn) is the set of all neighbors of σn. SiLU is the Sigmoid Linear Unit activation function. BN is the Batch Normalization operation. LN is the Linear transformation.

[0135] An illustration of the Sformer module is demonstrated in FIG. 3, (c). General graph neural network (GNN) modules from Matformer are used in some embodiments of the QCformer. In standard GNN models, message flow typically occurs from the edge level to the node level. However, with higher-dimensional components, message passing is extended from triangles to edges. The overall framework is illustrated in FIG. 3, (a). This generalized higher-dimensional message-passing scheme can be applied to any simplicial complex of any dimension, provided that boundary relations are properly defined.

[0136] Features on simplices of different dimensions are updated in distinct modules. In general, the feature vectors are updated using an attention mechanism that aggregates information from neighbouring simplices and cofaces, where aggregating is performed across multiple layers corresponding to different simplex dimensions. There may be any desired number of Node-wise Sformer layers - presently, there are five. Then the next highest dimensional simplices (lowest + 2) send messages to the lowest-dimensional simplices. Presently, higher-dimensional messages flow from 2-dimensional simplices to 0-dimensional simplices in a simultaneous process across two layers. Each layer includes an Edge-wise Sformer (306) that receives messages from the 2-dimensional simplices, processes those messages and passes them to a Nodewise Sformer (308). Where higher-dimensional simplices are used, e.g. tetrahedrons, the messages from the tetrahedron simplices (3-dimensional) will pass to Triangle-wise Sformers, from the Triangle-wise Sformers to the Edgewise Sformers, and from the Edge-wise Sformers to the Node-wise Sformers, to update feature vectors of vertices.

[0137] This approach prioritizes a broader receptive field for 0-dimensional simplices while preserving higher-dimensional information to prevent its loss through successive message-passing layers. As illustrated in FIG. 3, the 0-simplices (vertices) are first updated by Node-wise Sformer layer(s) (304) that take feature vectors of 0-simplices and 1-simplices as inputs. Subsequently, the 0-simplices are updated by simultaneous Edge-wise Transformer layer(s) (306), where Edge-wise Sformer layers process 1-simplices and 2-simplices as inputs, along with the updated feature vectors produces by the Node-wise Sformer layers (304). In this message passing scheme, the attention mechanism aggregates messages from an n-simplex's neighbours (n-simplices) and cofaces ( (n + l)-simplices). The updated feature vectors for the nodes and edges, can have large numbers of features. This issue is exacerbated for embodiments where feature vectors for higher-dimensional simplices - e.g., 3-dimensional simplices - are considered. To reduce dimensionality, average pooling layers (310), (312) are applied to the updated feature vectors. The average, updated feature vectors are fed to a MLP layer 314 for material property prediction. The MLP layer (314) projects simplices of all dimensions into the same hidden dimension, for prediction.

[0138] Experiments

[0139] The present framework was test on two benchmark datasets: Materials Project and JARVIS. Specifically, the 2018.6.1 version of the Materials Project dataset and the 2021.8.18 version of the JARVIS-DFT dataset were used. For the Materials Project dataset, the present model was evaluated on four crystal property prediction tasks, including Formation Energy, Band Gap, Bulk Moduli and Shear Moduli. For the JARVIS dataset, we consider five crystal property prediction tasks, including Formation Energy, Total Energy, Bandgap(OPT), Bandgap(MBJ) and Ehull. Matformer was again followed and the same training, validation and test sets were used for all the prediction tasks. The mean absolute error (MAE) is used to qualify the model performance.

[0140] The Materials Project dataset. We first evaluate our QCformer model using the Materials Project 2018.6.1 dataset, which contains 69,239 crystals - four crystal properties are considered, including Formation Energy, Band Gap, Bulk Moduli, and Shear Moduli. The tasks of Formation Energy and Band Gap share the same training, validation, and test set, including 60,000, 5,000, and 4,239 crystals, respectively; the tasks of Bulk Moduli and Shear Moduli use the same training, validation, and test set, including 4,664, 393 and 393 crystals, respectively.

[0141] The results, summarized in Table 1, show that QCformer outperforms all other models on three out of four tasks and achieves performance comparable to the second-best model on the Formation Energy task. Specifically, for Bulk Moduli and Shear Moduli, QCformer reduces the MAE of the second-best model by 13.16% and 7.81%, respectively. Moreover, QCformer achieves the best performance on Bulk Moduli and Shear Moduli with 4,664 training samples, and on Formation Energy and Band Gap with 60,000 training samples. This demonstrates the model's robustness across prediction tasks involving datasets of varying sizes.

[0142] Table 1: Model performance (MAE) comparison between QCformer and existing models on the Materials Project dataset. The units are eV / atom, eV, log (GPa), log (GPa) for Formation Energy, Band Gap, Bulk Moduli and Shear Moduli, respectively.

[0143]

[0144] The QCformer model was also evaluated on the JARVIS 2021.8.18 dataset, which comprises 55,722 crystals. Five properties were considered, including Formation Energy, Total Energy, Bandgap(OPT), Bandgap(MBJ) and Ehull. The tasks of Formation Energy, Total Energy, and Bandgap(OPT) use the same training, validation, and test set, including 44,578,5,572, and 5,572 crystals; the task of Bandgap(MBJ) uses 14,537, 1,817, and 1,817 crystals for training, validation, and test set, respectively. The results, presented in Table 2, demonstrate that QCformer outperforms all other models on the prediction tasks of Total Energy, Bandgap(OPT), and Bandgap(MBJ). Notably, QCformer reduces the MAE of the second-best model by 7.38% for Bandgap(OPT) and 7.69% for Bandgap(MBJ).

[0145] Table 2: Model performance (MAE) comparison between QCformer and existing models on the JARVIS dataset. The units are eV / atom, eV / atom, eV, eV, eV for Formation Energy, Total Energy, Bandgap(OPT), Bandgap(MBJ) and Ehull, respectively.

[0146]

[0147] The QCformer model was also used for perovskite property prediction. For this purpose, QCformer was pretrained using 55,722 data points from the JARVIS dataset on the Bandgap (OPT) task to generate useful graph embeddings. The pretrained model was then applied to downstream tasks for training on, and predicting, hybrid organic-inorganic perovskites bandgap. Performance was evaluated on two datasets, one being hybrid organic-inorganic perovskites (HOIP) and the other being 2D hybrid organic-inorganic perovskites (HOIP2D) proposed by Laboratory of New Materials for Solar Energetics (NMSE). The HOIP dataset contains 1,346 hybrid organic-inorganic perovskites, featuring 16 organic cations, 3 group-IV cations, and 4 halide anions. While both the HSE06 bandgap and the GGArPW86 bandgap are reported in the dataset, only the HSE06 bandgap is presently evaluated as it is closer to the true bandgap of the materials. A ten-fold cross-validation protocol is used for training and testing and the coefficient of determination (COD), pearson correlation coefficient (PCC), mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE) were provided for comparison with other baseline models. The HOIP2D dataset contains 849 hybrid organic-inorganic two-dimensional (2D) perovskites, containing 180 organic cations, 10 metals and 3 halide anions. Within this collection, 753 compounds are equipped with DFT-based bandgap values, while 238 compounds possess experimental bandgap values. Specially, only 716 compounds were used with DFT-based bandgap values, as supported by the unit cell representation from the pymatgen package. To ensure a fair comparison, QCformer was trained on a randomly selected subset of 624 data points (out of 716 compounds), repeating the subsampling procedure five times for each run to mitigate randomness. A five-fold cross-validation protocol was used for training and testing.

[0148] The results in Table 3 show that the present framework not only outperforms traditional machine-learning methods, but also achieves better accuracy than contemporary GNN approaches. For comparison, TPD-GBT, PRCa-GBT and PHa-GBT (67) are all gradient-boosted tree (GBT) models, each enhanced by different descriptor sets-traditional perovskite descriptors (TPD), persistent Ricci curvature (PRC), or persistent homology (PH), respectively. While incorporating topological features such as PRC and PH already improves predictive performance over purely descriptor-based machine learning (ML), GNN methods typically yield even better results. The present approach utilizes topological representations within simplicial-complex neural networks, providing further gains beyond both GBT-based and GNN-based techniques. In particular, a 30.59% improvement in MAE is observed over MEGNet, accompanied by a COD of 0.9829 and a PCC of 0.9916. These findings underscore the effectiveness of merging topological insights with neural architectures for accurately modeling bandgap properties.

[0149] Table 3: Model performance comparison between QCformer and existing models on the HOIP dataset. The comparison includes machine learning based models (TPD-GBT, PRCaGBT and PHa- GBT ), and GNN models (SchNet and MEGNet). The units are eV, eV2and eV for MAE, MSE and RMSE.

[0150]

[0151] As summarized in Table 4, QCformer outperforms a range of GNN-based methods, including GCN, ECCN, CGCNN, TFGNN, SIGNNA and SIGNNAC. Among these, SIGNNAcattains the best performance, primarily by separating organic and inorganic sub-structures. By incorporating topological features, the present method achieves a MAE of 0.0754, approximately 9.2% lower than SIGNNAc. However, in general, GNN methods appear limited by the relatively small training set, which negatively impacts their performance compared to GBT models.

[0152] Table 4: Model performance comparison between QCformer and existing models on the HOIP2D dataset. The comparison includes machine learning based models (SOAP-KRR, SOAP-MLM1, and GDA(IF)-GBT), and GNN models (GCN, ECCN, CGCNN, TFGNN, SIGNNA and SIGNNAc). The second column shows the numbers of materials involved in different experiments. The units are eV and eV for MAE and RMSE.

[0153]

[0154] QCformer has also been used for predicting unknown bandgap labels for 2D perovskites from the HOIP2D dataset. QCformer was trained on 624 data points from HOIP2D with known bandgaps, then tested on five unlabelled compounds indexed 110,309,315,363, and 452. To provide a prediction benchmark, GGA-PBE-based DFT calculations were performed on these same compounds. Across the five compounds, QCformer achieves a MAE of 0.3849 eV as shown in Table 5. Although this MAE is higher than the result in Table 4, this is expected, as a smaller dataset can lead to larger average prediction errors. Table 5: Comparison of DFT and predicted bandgaps for five new compounds. The first column and second column show the indices and chemical formulas. The third column presents calculated bandgaps by GGA-PBE-based DFT method. The fourth column showcase the predicted bandgaps by QCformer.

[0155] No. Formula DFT bg (eV) QCformer bg (eV) 110 [(C6H11)PH3]2SnI41.3540 2.2435

[0156] 309 [C2H5NH3]2FeCl42.7002 2.5898

[0157] 315 Cs3Sb2I91.5450 1.7782

[0158] 363 [CH3NH3]2FeCl43.1661 3.1757

[0159] 452 Tl3Bi3I92.3780 1.6962

[0160] This is further illustrated in FIG. 10, showing close alignment of QCformer predictions with DFT bandgaps across all five materials. For instance, QCformer accurately predicts the higher bandgap of material 363, with a minimal deviation from GGA-PBE values. Similarly, the predicted bandgap of material 315 aligns closely with the r2SCAN bandgap, suggesting that QCformer can capture trends observed across multiple computational methods.

[0161] FIG. 11 illustrates a system (1100) for performing the method (100). The system includes memory (1102), that may be any non-transitory storage medium, for storing computer program instructions (1104) that are executable to run the method (100).

[0162] One or more processors (1106) extract the instructions (1104) from memory (1102) to run method (100). In particular, execution of the instructions causes the quotient complex modeller (1108) to receive an input, any desired format including from material libraries, comprising a crystal structure for which material properties are to be predicted. The quotient complex modeller (1108) models pairwise and many-body interactions as discussed in relation to step (102), using simplices of a plurality of dimensions. The quotient complex modeller (1108) also models periodic material properties by applying a quotient operation over the simplices. The quotient complex modeller (1108) then generates a quotient complex (QC) from the modelled interactions and periodic material properties.

[0163] Once the QC has been generated, the transformer module (1110) uses the feature generator (1112) to generate a feature vector for each simplex in the QC. The feature vectors are then updated based on feature vectors of one or both of neighbouring said simplexes in the QC and cofaces of the respective simplex, using a message passing scheme executed by the Node-wise Sformer module (1114) and the Edge-wise Sformer module (1116). A prediction layer (1118) takes the QC with updated feature vectors and processes it using a simplex-based transformer to predict one or more material properties of the crystal structure.

[0164] It will be appreciated that particular modules are illustrated as being implemented in hardware. However, any such modules can also be implemented in software or a combination of software and hardware. Moreover, additional or alternative system components may be used as necessary.

[0165] The above discussion demonstrates an efficient framework for incorporating topological and periodic atomic interactions from crystalline materials, into a QC model for efficient material property prediction.

[0166] It will be appreciated that many further modifications and permutations of various aspects of the described embodiments are possible. Accordingly, the described aspects are intended to embrace all such alterations, modifications, and variations that fall within the spirit and scope of the appended claims.

[0167] Throughout this specification and the claims which follow, unless the context requires otherwise, the word "comprise", and variations such as "comprises" and "comprising", will be understood to imply the inclusion of a stated integer or step or group of integers or steps but not the exclusion of any other integer or step or group of integers or steps.

Claims

Claims1. A material property prediction method comprising:modelling a periodic structure as a quotient complex (QC), the quotient complex modelling:pairwise and many-body interactions using simplices of plurality of dimensions; andperiodic material properties by applying a quotient operation over the simplices;generating a feature vector for each simplex in the QC;updating the feature vector for each simplex in the QC based on feature vectors of one or both of neighbouring simplexes in the QC and cofaces of the respective simplex, using a message passing scheme; and processing the QC with updated feature vectors using a simplex transformer to predict one or more material properties of the periodic structure.

2. The method of claim 1, wherein updating the feature vector for each simplex in the QC comprises, for each simplex in the QC:computing an attention coefficient vector between the respective simplex and each other simplex in the QC;computing a message based on the attention coefficient; and updating the feature vector based on the message.

3. The method of claim 1 or 2, wherein each simplex comprises a vertex, edge or simplex of higher-dimension than an edge, and applying the quotient operation comprises merging equivalent vertices, edges, and higherdimensional simplices based on periodic equivalence relations defined over the periodic structure.

4. The method of claim 3, wherein said feature vector for each simplex is generated using one-hot encoding for vertices, and radial basis functions(RBFs) for other simplices.

5. The method of any one of claims 1 to 4, wherein said message passing scheme updates feature vectors of 0-simplices through a sequence of five Node Transformer layers followed by two simultaneous Edge-wise Transformer layers, with messages flowing from higher-dimensional simplices to lower-dimensional simplices.

6. The method of claim 5, wherein the Edge-wise Transformer layers process simplices other than 0-simplices, and an attention mechanism of each Edge-wise Transformer layer aggregates messages from neighbouring simplices and cofaces of respective simplices.

7. The method of any one of claims 1 to 6, wherein the feature vector for each simplex includes one or more attributes selected from the group consisting of atomic properties, bond lengths, angles between bonds, and geometric relationships within the periodic structure.

8. The method of any one of claims 1 to 5, wherein updating the feature vector for each simplex in the QC using the message passing scheme, comprises updating the feature vectors using an attention mechanism that aggregates information from neighbouring simplices and cofaces, where aggregation is performed across multiple layers corresponding to different simplex dimensions.

9. The method of claim 8, wherein the attention mechanism operates such that n-dimensional simplices, of the simplices of the QC, firstly receive messages exclusively from (n+1)-dimensional simplices of the simplices of the QC, using an initial set of layers of the simplex transformer, followed by simultaneous updates from (n+2)-dimensional simplices in subsequent layers.

10. The method of claim 9, wherein the n-dimensional simplices are 0-dimensional simplices.

11. The method of claim 4, wherein attributes of the feature vector for each simplex in the QC, other than 0-dimensional simplices, include encoded geometric attributes of the periodic structure, derived from the RBFs.

12. The method of claim 3, 4, or 11, wherein the periodic structure is a crystal structure assumed equivalent to an infinite crystal graph, the equivalence relations being defined based on periodic patterns inherent in the crystal structure, and the quotient operation reducing the infinite crystal graph to a finite representation.

13. The method of claim 6 or 8, wherein aggregating comprises aggregating feature vectors from neighbouring simplices and cofaces using a scaled dotproduct attention mechanism.

14. The method of claim 13, wherein the scaled dot-product attention mechanism computes attention coefficients between an n-simplex and its neighbours.

15. The method of any one of claims 1 to 14, wherein the simplex transformer processes the updated feature vectors through multiple Simplex Transformer (Sformer) blocks, each Sformer block being configured to independently update embeddings for simplexes, in the QC, of different dimensionality.

16. A system for predicting material properties of crystal structure materials, comprising:memory;at least one processor;a quotient complex modeller; anda transformer module,wherein the memory stores instructions that, when executed by the at least one processor, cause the:quotient complex modeller to receive input comprising a periodic structure, model pairwise and many-body interactions using simplices of plurality of dimensions, model periodic material properties by applying a quotient operation over the simplices, and generate a quotient complex (QC) from the modelled interactions and periodic material properties;transformer module to:generate a feature vector for each simplex in the QC;update the feature vector for each simplex in the QC based on feature vectors of one or both of neighbouring said simplexes in the QC and cofaces of the respective simplex, using a message passing scheme; andprocess the QC with updated feature vectors using a simplex transformer to predict one or more material properties of the periodic structure.

17. The system of claim 16, wherein the transformer module is configured to update the feature vector for each simplex in the QC, by:computing an attention coefficient vector between the respective simplex and each other simplex in the QC;computing a message based on the attention coefficient; and updating the feature vector based on the message.

18. The system of claim 17 or 18, wherein each simplex comprises a vertex, edge or simplex of higher-dimension than an edge, and applying the quotient operation comprises merging equivalent vertices, edges, and higherdimensional simplices based on periodic equivalence relations defined over the periodic structure.

19. The system of any one of claims 16 to 18, wherein the transformer module uses the message passing scheme to update feature vectors of 0-simplices through a sequence of five Node Transformer layers followed by two simultaneous Edge-wise Transformer layers, with messages flowing from higher-dimensional simplices to lower-dimensional simplices.

20. The system of claim 19, wherein the Edge-wise Transformer layers process simplices other than 0-simplices, and an attention mechanism of each Edge-wise Transformer layer aggregates messages from neighbouring simplices and cofaces of respective simplices.

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