A particle orbit discovery method based on a graph neural network cellular automaton

Abstract

A particle track discovery method based on graph neural network cellular automaton, which utilizes a high-energy physics detector to record a plurality of particle hit data generated in a particle collision event, including the position, direction and momentum of the particle at its hit point; using a plurality of particle hit data generated in multiple particle collision events to supervise the training of a graph neural network, and equivalently calculating a generated graph neural cellular automaton; adding the graph neural cellular automaton conversion rule into a conversion rule library; selecting a plurality of graph neural cellular automaton conversion rules from the conversion rule library, equivalently calculating a corresponding plurality of graph neural networks; applying the plurality of graph neural networks in step three to a new particle collision experiment, inputting a plurality of groups of measurement values of the same particle obtained in the new particle collision experiment into the plurality of graph neural networks, obtaining a plurality of different predicted tracks, and discovering the real track of the particle from the different predicted tracks according to the known physical properties of the particle.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of high-energy physics particle measurement and the application field of deep learning graph neural network and cellular automata, and particularly relates to a particle orbit discovery method based on a graph neural network cellular automaton. BACKGROUND

[0002] In the Large Hadron Collider, two general-purpose detectors, A Toroidal LHC ApparatuS (ATLAS) and Compact Muon Solenoid (CMS), are used to analyze the large number of particles produced when the accelerator collides. Two beams of high-energy particles travel at close to the speed of light in opposite directions before colliding. Both beams are in a state of ultra-high vacuum, and a strong magnetic field causes them to run around the accelerator ring. The Large Hadron Collider uses thousands of different types of magnets to direct the particle beams around the accelerator. Among them, 1232 dipole magnets are used to bend the particle beams, and 392 quadrupole magnets are used to concentrate the particle flow. By squeezing the particles through these magnets, the probability of successful collision is increased.

[0003] The pattern recognition problem in high-energy physics is different from the machine learning application in computer vision. The particle reconstruction algorithm identifies the kinematic properties of the particles produced in high-energy collisions. The problem of reconstructing the charged particle orbit in a high-energy physics detector has the characteristics of high dimensionality, high sparsity, and complex geometric layout.

[0004] Under existing experimental conditions, some measurements are noise, and some are particles with too low energy. In order to find a set of possible candidate orbits, many hypotheses must be explored, and finding the orbit is usually a tedious and time-consuming process. Computational speed is an important issue that also determines the choice of algorithm.

[0005] There are two main existing technologies for charged particle orbit discovery: conformal mapping and Hough transform.

[0006] The main shortcomings of the two technologies are: slow computing speed, low accuracy, and low prediction accuracy. In order to reduce the computation time, the orbit discovery usually uses a known model of the charged particle orbit, or a simplified version of it. SUMMARY

[0007] In order to overcome the shortcomings of the above-mentioned existing technologies, the purpose of the present application is to provide a particle orbit discovery method based on a graph neural network cellular automaton, which combines deep learning graph neural network and cellular automaton to solve the recognition and classification problems generated by the orbit discovery of a large number of particle collisions in the prior art, and to improve the accuracy of particle orbit discovery.

[0008] To achieve the above object, the technical scheme adopted by the present application is:

[0009] A particle orbit discovery method based on a graph neural network cellular automaton, comprising the following steps:

[0010] Step one, data generation: using a high-energy physics detector to record a plurality of particle hit data generated in a particle collision event, including the position, direction and momentum of the particle at its hit point;

[0011] Step two, training a graph neural network: using a plurality of particle hit data generated in a plurality of particle collision events to supervise the training of a graph neural network;

[0012] Step three, introducing a graph cellular automaton: generating a graph neural cellular automaton by equivalent calculation of the graph neural network in step two; adding the graph neural cellular automaton conversion rule to the conversion rule library; selecting a plurality of graph neural cellular automaton conversion rules from the conversion rule library to generate a plurality of corresponding graph neural networks by equivalent calculation;

[0013] Step four, orbit discovery: applying the plurality of graph neural networks in step three to a new particle collision experiment, inputting a plurality of measurement values of the same particle obtained in the new particle collision experiment into the plurality of graph neural networks to obtain a plurality of different predicted orbits, and discovering the real orbit of the particle from the different predicted orbits according to the known physical properties of the particle, and inferring the new physical properties of the particle.

[0014] In one embodiment, the step one comprises:

[0015] First stage: placing a group of high-energy physics detectors near the particle collision area;

[0016] Second stage: the charged particles generated in the collision will ionize the sensitive layer of the material in the collision accelerator tube when they leave the collision area, and will generate a few particle position measurement values along the orbit of each particle; the high-energy physics detector records the position of all particles at certain time intervals, and all particles measured at the same time will form a circle, called a detector layer; periodic measurement forms a plurality of detector layers, and all detector layers form a plurality of concentric circles; the detector layers measured at adjacent time intervals are defined as adjacent layers.

[0017] In one embodiment, in the first stage, the high-energy physics detector is immersed in a strong magnetic field of a curved orbit to minimize the amount of irrelevant substances present in the high-energy physics detector, and the momentum component measurement method uses the principle that the curvature is proportional to the momentum component transverse to the magnetic field.

[0018] In one embodiment, in the step two, the particle hit data generated in the particle collision event is represented by the graph structure connected by the hit points; if the hit points meet the graph neural network judgment rule, the hit points are connected with the hit points of the adjacent layer; the hit point refers to the measurement record of the position of the current particle by the high-energy physics detector at a certain time interval, and the particle hit data refers to the measurement record of the position of all particles by the high-energy physics detector at the same time; in the graph neural network, the graph node corresponds to the hit point, and the edge corresponds to the connection of the hit points.

[0019] In one embodiment, in the step two, the graph neural network is composed of an input conversion layer and a plurality of cyclically alternating edge networks and node networks, and the final output layer is an edge network; the input of the graph neural network is node feature and connectivity rule, and the node feature refers to hit coordinates in three-dimensional space; the graph neural network performs two different tasks:

[0020] 1) a binary hit classification model, which learns to identify a certain local track in the graph by classifying the graph nodes.

[0021] 2) a segment classifier model, which learns to identify multiple tracks at a time by classifying the edges in the graph;

[0022] Among them, the edge network uses the features of the start node and the end node to calculate the weight value of each edge of the graph; the node network respectively uses the nodes on the last detector layer and the next detector layer, and inputs the current feature of the node and the edge weight value between the nodes into an aggregation function to calculate the new feature of each node.

[0023] In one embodiment, the edge network and the node network are both implemented by multi-layer perceptron (MLP), each MLP has two hidden layers and a hyperbolic tangent activation function.

[0024] In one embodiment, in the step three, the anisotropy in the graph cellular automaton is regarded as a dependency relationship of the conversion rule, and the dependency relationship represents a specific relationship existing between the cell and the neighbor, and the specific relationship refers to taking the union of the current state of node i and the current state of all neighbor nodes j of node i, and taking the union of the edge weight between node i and all neighbor nodes j of node i as the next state of node i, i.e. i , so that different neighbor nodes j of i can be treated differently according to the edge weight of node j relative to node i.

[0025] In one embodiment, the state conversion function of the graph cellular automaton is implemented by using the graph neural network to obtain the graph neural cellular automaton, wherein:

[0026] First, the state of the graph cellular automaton is interpreted as a graph signal when the graph neural network is configured as d = d'; d is the dimension of the vector attribute function h i and d' is the dimension of the graph neural network message passing scheme function h' i ;

[0027] Second, the isotropic or anisotropic transition rule is modeled with a graph neural network;

[0028] Finally, the graph neural cellular automaton is implemented by stacking multiple message passing layers or redefining the neighbor function to include higher order neighbors, thus realizing a cellular automaton rule with a larger range of neighbors.

[0029] In one embodiment, the graph cellular automaton is implemented with M state spaces, i.e., the equivalent computation of the graph neural network and the graph neural cellular automaton, which relies on two neural computing blocks:

[0030] The first neural computing block is two convolutional layers with 1x1 filters to compute the one-hot encoding of the state;

[0031] The second neural computing block is one convolutional layer with 3x3 filters to match all possible input patterns of the transition rule and compute the corresponding output;

[0032] The transition rule design process is divided into two stages:

[0033] Stage 1: the first neural computing block remains unchanged;

[0034] Consider a graph cellular automaton with M cells and a state space and the maximum node degree d max ≤ M, the first neural computing block is a transition applied to each node independently of its neighbor nodes;

[0035] Stage 2: the second neural computing block remains unchanged;

[0036] Each cell has d max neighbor nodes, which are identified by a unique one-hot vector edge attribute, and the pattern matching mechanism is implemented with a weight matrix of shape d max dimension, which is used to compute the message from each neighbor node; extended to continuous state space, the expressiveness of the architecture described by the graph neural network is sufficient to represent any transition function of the graph cellular automaton with continuous state space.

[0037] In one embodiment, in the graph cellular automaton, the value of a particular position is the sum modulo of the values of its two adjacent positions at the previous time step:

[0038]

[0039] s t (site) is the value of the point site at time step t, represents a modulo addition operation, The front and rear elements always have symmetry;

[0040] According to the evolution of the graph cellular automaton rule, the superposition principle is satisfied:

[0041]

[0042] Compared with the prior art, the beneficial effects of the present application are:

[0043] Firstly, the present application uses a graph neural cellular automaton to automatically learn the conversion rule in the particle collision experiment, instead of through artificial design, and opens up the application limit of the graph neural cellular automaton through the graph neural network, improves the efficiency of orbit discovery, and can better understand the characteristics of the particle orbit.

[0044] Secondly, through the graph neural network representation of the graph neural cellular automaton as a bridge, more conversion rules in the conversion rule library of the graph neural cellular automaton can be applied in the particle collision experiment orbit discovery process, providing multiple hypotheses and predictions for the discovery of new particle physics characteristics. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 The flowchart of the present application.

[0046] Figure 2 The particle orbit discovery schematic diagram of the present application.

[0047] Figure 3 The graph structure representation of the orbit hit data of the present application.

[0048] Figure 4 The orbit discovery graph neural network proposed by the present application. DETAILED DESCRIPTION

[0049] The embodiments of the present application will be described in detail below in combination with the drawings and examples.

[0050] The graph neural network (GNN) is a graph data structure composed of a group of node elements with feature attributes and edges connecting pairs of nodes. Physical data generated by particles, such as hit data in a detector, can be represented by a graph data structure such as GNN.

[0051] Cellular Automata (CA) is a class of computational models that study complexity, whose core is the dynamic characteristics generated by the local interaction of cells arranged in regular cells. Cellular Automata is a discrete computational model, which is composed of regular cells, each of which is associated with a state and a transition rule.

[0052] The transition rule is usually defined as a function that updates the state of each cell to its current state and the state of adjacent cells.

[0053] Even if simple transition rules and low-dimensional cells are used, cellular automata can generate very complex patterns by repeatedly applying transition rules. Cellular automata can construct a model of some real-world phenomenon, such as the collision orbits of charged particles.

[0054] The present invention uses a generalized version of cellular automata, called Graph Cellular Automata (GCA), which relaxes the assumption of cell structure, replaces it with a graph data structure, and allows cells to have an arbitrary and variable number of neighbors.

[0055] By observing the similar features between convolutional neural networks (CNN) and graph neural networks (GNN), cellular automata (CA) and graph cellular automata (GCA), the combination of GNN and GCA realizes an application: using GNN to learn a desired GCA transition rule, called Graph Neural Cellular Automata (GNCA). By applying GNN to construct GCA, the present invention can discover unknown rules, so that the present invention can solve new emerging distributed computing tasks, such as global traffic prediction problems and particle collision orbit prediction problems.

[0056] In particular, with reference to Figure 1 The present invention is a particle orbit discovery method based on graph neural network cellular automata, characterized by the following steps:

[0057] Step 1, generate data: use high-energy physics detectors to record multiple particle hit data generated in particle collision events, including the position, direction and momentum of the particles at their hit points.

[0058] In the present invention, the physical concepts related to particle collisions are as follows:

[0059] The basic unit of collision is particle group, which is composed of multiple particles. After collision, two particle groups generate multiple new particles, which move outward from the center and ionize the material sensitive layer in the collision accelerator tube. During the outward movement of all particles, the high-energy physics detector records the position of all particles at certain time intervals by measuring and analyzing the material sensitive layer. All particles measured at the same time form a circle, which is called a detector layer. Therefore, periodic measurement forms multiple layers, and all layers form multiple concentric circles. The detector layer measured at adjacent time is defined as adjacent layer.

[0060] Hit point refers to the measurement and record of the current particle position by the high-energy physics detector at certain time intervals.

[0061] The high-energy physics detector can measure and record the position of all particles at the same time, which is defined as particle hit data.

[0062] The task of orbit reconstruction is divided into two sub-tasks: orbit discovery and orbit fitting.

[0063] The orbit is the actual motion orbit of the current particle. Because it is impossible to determine which hit point is measured by the current particle, it is impossible to determine the orbit of the current particle, so orbit discovery is needed.

[0064] The subset is classification, which is defined as a set of measurements of the same particle in the present invention. Orbit discovery is a classification problem, that is, all measurement value sets in the detector are divided into different classifications, and each classification contains measurement values from the same particle.

[0065] Because in the collision:

[0066] 1. Collision generates multiple particles;

[0067] 2. There are many particles of the same kind, but each particle only forms one orbit.

[0068] Therefore, the core of the orbit discovery task is to associate the hit point belonging to the same particle with its own orbit. Orbit discovery in the present invention is a pattern recognition or classification problem, and the purpose is to divide the measurement value set in the detector into subsets containing measurement values from the same particle.

[0069] The high-energy physics data analysis process in step one of the present invention should as accurately as possible estimate the particle dynamics characteristics generated in the collision event, such as the position, direction and momentum of the particle at its hit point. The specific process can be divided into two stages:

[0070] The first stage: a set of high-energy physics detectors are placed near the particle collision region; for example, to prevent the detector elements from interfering with the particle's trajectory, the high-energy physics detectors are immersed in a strong magnetic field with a curved trajectory to minimize the amount of extraneous matter present in the high-energy physics detectors, and the momentum component measurement method uses the principle that the curvature is proportional to the momentum component transverse to the magnetic field.

[0071] The second stage: the charged particles produced in the collision ionize the sensitive layers of the material in the collision region as they leave the collision region, and several particle position measurements are made along the trajectory of each particle.

[0072] The high-energy physics detectors record the positions of all particles at certain times, and all particles measured at the same time form a circle called a detector layer. Multiple detector layers are formed by periodic measurements, and all detector layers form multiple concentric circles. The detector layers measured at adjacent times are defined as adjacent layers.

[0073] General-purpose detectors such as ATLAS and CMS have a cylindrical space 40 meters long and 25 meters in diameter, with a very complex geometry, and the data generated is high-dimensional and sparse. Detector subsystems and algorithms are used to reconstruct different types and properties of particles produced in collisions. For example, the trajectory of a charged particle in a tracking detector is reconstructed from spatial point measurements (i.e., hits). There are particles and hits in each event. The fitted parameters of the trajectory, particularly the curvature, allow the momentum and charge of the particle to be measured.

[0074] Step two, training the graph neural network: use the multiple particle hit data generated in multiple particle collision events to supervise the training of the graph neural network.

[0075] Traditional particle trajectory finding methods use combinatorial search algorithms guided by Kalman filters. These algorithms are optimized for physical performance under Large Hadron Collider conditions, but are essentially sequential. Ideally, each particle would leave only one hit on each layer of the detector, and the trajectory would be an exact spiral, with precise coordinates. In reality, particles can leave multiple hits or no hits in a layer, the inhomogeneity of the magnetic field causes the arc to twist, particles can experience multiple scatterings, and measurements can have anisotropic uncertainties.

[0076] Figure 2 A simple particle trajectory finding problem is shown. Each colored curve is the trajectory of a charged particle in a constant magnetic field perpendicular to the observation plane. The solid circles are hit points left by the particle as it passes through the sensitive layers. The hollow circles are pseudo-noise hits, which are generated by unresolvable particles.

[0077] The track finding algorithm used in the present application is an improved algorithm based on the combinatorial track finder (CTF) of the combinatorial Kalman filter. The present application uses a graph structure representation of multiple particle hit data generated in a particle collision event by connecting hit points.

[0078] The graph structure representation of track hit data is as shown in Figure 3 If the hit points meet the rules determined by the graph neural network, they will be connected with the hit points of the adjacent layer.

[0079] In this representation, the present application can apply the graph neural network (GNN) from deep learning, and connect the hit points that may belong to the same track through pre-processing algorithms such as geometric constraints or Hough transform. The GNN learns this representation and solves the track finding task through the prediction of graph nodes (corresponding to hit points), edges (corresponding to the connection of hit points) or global state, so that the performance and robustness are balanced.

[0080] The GNN in the present application is composed of an input conversion layer and several cyclically alternating edge networks and node networks, and the system structure is as shown in Figure 4 It starts from an input conversion layer, followed by multiple alternating edge networks and node networks for cyclic iteration. In this case, the final output layer is an edge network, which makes it a segment classifier model. With each iteration of the neural network, this model propagates information through the graph, adaptively learns to strengthen important connections and weaken irrelevant connections.

[0081] The input of the graph neural network of the present application is the node feature and connectivity rule, and two different tasks can be performed:

[0082] 1. Binary hit classification model, which learns to identify a local track in the graph by classifying the graph nodes.

[0083] 2. Segment classifier model, which learns to identify multiple tracks at a time by classifying the edges (pairs of hit points) in the graph.

[0084] The two main network layers perform calculations locally in the graph. The edge network uses the features of the start node and the end node to calculate the weight value of each edge of the graph; the node network uses the node features of the previous detector layer and the next detector layer, respectively, and inputs the current node features and edge weight values between nodes into an aggregation function to calculate the new features of each node.

[0085] In the present application, the feature can be a set of state values of a node, which is a hit coordinate in a three-dimensional space, and the set of state values represents the physical information of the node and can be changed. Therefore, the current feature of the node is the state value at the current moment, and the new feature of the node is the state value at the next moment.

[0086] In the graph structure, each node represents a hit, and edges are constructed between pairs of hit points on adjacent layers, which can belong to the same track. By updating the node representation and edge weight values multiple times and using the learned edge weight values as an attention mechanism, the segment classifier model learns which edges actually connect hit points belonging to the same track. This method converts the clustering problem into edge classification by focusing on hit subgraphs belonging to the same track.

[0087] In the present application, the edge network and the node network are both implemented by a multi-layer perceptron (MLP), each MLP having two hidden layers and a hyperbolic tangent activation function.

[0088] When the static structure of the graph neural network is determined, the entire network needs a mechanism to dynamically update the nodes, i.e., the mechanism of updating the state of the nodes through the message passing layer. Formally, the l-th GNN layer is defined as:

[0089]

[0090] is the embedding representation of node v at the l-th layer; W (l) and b (l) are weight parameters; v is the current node, and u is the main node of the entire graph; is the set of local neighbor nodes of node v; Aggr(·) is an aggregation function; Activation(·) is a nonlinear activation function; Dropout(·) is a common dropout layer in deep learning; and BatchNorm(·) is a common batch normalization layer in deep learning.

[0091] In an embodiment of the present application, a graph structure representation of data is defined, a single detector measurement is used as a node, and a heuristic-based domain knowledge method is used to construct edges between nodes. The main task of the GNN is to associate the detector elements together by classifying the edges of the graph. Track discovery only considers tracks and impacts within the detector barrel region.

[0092] The construction of the graph is as follows: the nodes are hits of detector records, and the edges are hit connections between adjacent detector layers, which are made valid for tracks of transversely high-momentum particles by a pre-defined filter.

[0093] In the input graph, the node features are three cylindrical coordinates (r, Θ, z), while the edge features are the difference of the coordinates (Δφ, ΔΘ). If two hits come from the same orbit, the edge label value is 1, otherwise 0.

[0094] The GNN architecture has three components: an encoder that transforms the input node and edge features into their latent representations; a graph module that performs message passing to update the latent features; and an output module that computes the edge classification scores.

[0095] The GNN model used is based on the interaction network architecture. It is a GNN model based on stacked, repeated updates of interaction network layers.

[0096] The simplified representation of nodes and edges is as follows:

[0097] I t+1 = f R (o 1,t , o 2,t , r)

[0098] o 2,t+1 = f O (o 2,t , I t+1 )

[0099] In the single node case: f O is the object-centric function, o t is the input object state, t is the current time, and o t+1 is the output state at the next time.

[0100] In the two-node case: f R is the relation-centric function, o1 is the sender, o2 is the receiver, r is the relation attribute between nodes (e.g., edge), and I t+1 is the interaction influence value.

[0101] The core interaction network layer is defined as follows:

[0102] InteractionNetwork(G) = Φ o (aggr(G, X, Φ R (Mar(G))))

[0103] G = <O, R> is the formal extension of I t+1 to larger and more complex systems and represented with a graph data structure; O is the set of objects / nodes, represented by ; R is the set of edges / relations, represented by o i represents the i-th sender; o j represents the j-th receiver; r krepresents the kth relational attribute.

[0104] The specific functions of the interaction network are unfolded as follows:

[0105]

[0106]

[0107] f R (a k ) = c k

[0108] f O (b j ) = d j

[0109]

[0110]

[0111] Mar represents a marshalling function that rearranges objects and relations into interactions; a k = <o i , o j , r k > ∈ A represents a set of marshalling functions, one for each relation; Φ R represents a relational model that applies f R to each a k to predict the impact of each interaction; c k ∈ C represents the interaction impact value; aggr represents an aggregation function that aggregates all interaction impact values c k ; Φ O represents an object model that applies f O to each b j to predict how the interaction affects the target state; d j ∈ D is the return value of φ O .

[0112] This interaction network can predict the state evolution of dynamic systems.

[0113] In the GNN model architecture, the interaction network layer operates on the initial latent features H0connected to the current features H i-1 After 8 iterations, the output FC network takes the last latent feature H8to generate classification scores for each edge.

[0114] The encoder layer uses two fully connected 2-layer networks to transform node and edge features, respectively. The initial latent features of nodes and edges are collectively referred to as H0. The graph module is applied recursively to the latent features. In each iteration i, the initial features H0are connected to the current features Hi The short connections can improve the performance of the model.

[0115] The graph module also uses two fully connected 2-layer networks, one to compute the update edge features and the other to aggregate incoming edge features to compute the update node features.

[0116] After N iterations of the graph module, the output module takes the last latent feature H N and uses a 2-layer fully connected network to classify and score each edge.

[0117] All fully connected layers have a hidden layer size of 128 with ReLU function as the activation function, except the last layer of the output module uses sigmoid function activation.

[0118] Step three, introduce graph cellular automata: the graph neural network in step two is equivalent to calculation (represented by GNN or described GNCA conversion rule) to generate graph neural cellular automata; the graph neural cellular automata conversion rule is added to the conversion rule library; a plurality of graph neural cellular automata conversion rules are selected from the conversion rule library, and the corresponding graph neural network is generated by equivalent calculation.

[0119] In the application, the equivalent calculation of GNN and GNCA is equivalent to building a bridge between CA conversion rules and high-energy physics orbit discovery, and its role is bidirectional: the orbit discovery data in the existing experiment is represented by GNN, and then equivalent to GNCA, and finally added to the conversion rule library; in turn, the similar conversion rules in other fields are converted into GNCA, and then represented by GNN, and finally used in orbit discovery, the purpose is to discover previously undiscovered or misidentified orbits, so as to discover new particles.

[0120] Since GNN is only a neural network, it is obtained by training specific domain data. But the GNCA conversion rule does not only come from the application of charged particle orbit discovery, the GNCA conversion rule obtained from other fields can also be equivalent to a GNN and applied to charged particle orbit discovery, further improving the probability of discovering previously undiscovered or misidentified orbits, and also improving the probability of discovering new particles.

[0121] Specifically, in this step, a general architecture of graph neural network GNN is used to learn various transition rules of any kind of graph cellular automata GCA with finite and discrete state space, which can mimic one-dimensional and two-dimensional cellular automata with chaotic or complex dynamics. The present application is mainly based on two principles: 1. Using GCA transition rules can realize any CA transition rule; 2. Any GCA transition rule can be learned by GNN using gradient descent method. Since the convolutional neural network can be interpreted as a special case of GNN with fixed-size neighborhood and anisotropic filters, the CNN can be generally generalized to GNN, which extends the method of learning traditional CA transition rules using CNN. The modules are briefly introduced as follows:

[0122] Cellular automata (CA) is a class of computational models that study complexity, whose core is the dynamic characteristics generated by the local interaction of cells arranged in regular cells. Cellular automata is a discrete computational model, which is composed of regular cells, each of which is associated with a state and a transition rule. Even with simple transition rules and low-dimensional cells, cellular automata can produce very complex patterns by repeatedly applying the transition rules.

[0123] The basic elements of cellular automata are as follows:

[0124] Space: the area where cells are located in space, which can be one-dimensional, two-dimensional or multi-dimensional.

[0125] State set: can be two states, represented by 0, 1, black and white; or multiple states, such as different colors.

[0126] Neighbor: exists around a cell and can affect the state of the cell at the next time.

[0127] Evolution rule / transition rule: determines the dynamic function of the state of the cell at the next time according to the state of the cell and its neighbor cells, or the state transition equation. The transition rule is usually defined as a function that updates the state of each cell based on its current state and the state of neighboring cells.

[0128] The most core attribute in cellular automata is the transition rule, which can be simply understood as the law or function of the evolution and development of things. Abstractly defined as a dynamic function or state transition equation that determines the state of the cell at the next time according to the state of the cell and its neighbor cells.

[0129] Early transition rules were set by humans or summarized from experience. Later, many transition rules were abstracted from other fields (such as biology, chemistry, traffic, social phenomena, etc.).

[0130] Many transition rules have been set and discovered in the field of cellular automata for many years, forming a transition rule library. The more complex transition rules in the transition rule library are defined as complex patterns, which are used to explain complex system phenomena.

[0131] A graph cellular automaton (GCA) is a four-tuple is a graph data structure, where nodes are represented by N (|N| = n), n is the number of nodes, and edges are represented by E ∈ N × N. i represents the current node, and j represents the adjacent node connected to the current node. The state set is represented by , represents the current state of the node. is a function of aggregating neighbor node information, represented by , represents the local transition rule.

[0132] The definition of the transition rule is as follows:

[0133]

[0134] s′ i is the next state of the node.

[0135] The meaning of the transition rule is to take the union set of the current state of node i and the current state of all neighbor nodes j of i as the next state of node i, i.e. i .

[0136] In GCA, anisotropy can be obtained by enumerating transition rules.

[0137] More generally, the present application can assign an attribute to each edge (i, j) ∈ E as encoding, for example, assigning an attribute of direction, distance or unique identification to the edge of node i with respect to node j. Any arbitrary attribute associated with the edge can also be considered, which represents an abstract relationship between any two adjacent cells.

[0138] In short, the present application can regard anisotropy in GCA as a kind of dependence relationship on the transition rule, which represents a specific relationship existing between the cell and the neighbor:

[0139]

[0140] This specific relationship is an extension of the transition rule in the previous paragraph, which means not only taking the union set of the current state of node i and the current state of all neighbor nodes j of i, but also taking the union set of the edge weight between node i and node j as the next state of node i, i.e.i This has the advantage that different neighbor nodes j of i can be treated differently depending on their edge weight relative to node i.

[0141] Graph Neural Networks (GNN) is a set of models designed to process graph-structured data. Their core functionality can be described in terms of a message passing scheme. Let denote the vector attribute function associated with the i-th node of the graph, and let denote the edge attribute defined above.

[0142] A GNN message passing scheme is a function that computes a vector

[0143]

[0144] where Φ is a message function, PI is a permutation invariant operation that aggregates a set of incoming messages, and η is an update function.

[0145] When Φ, PI and η are differentiable operations, message passing layers can be stacked and learned by backpropagation and stochastic gradient descent.

[0146] The message passing architecture is as follows:

[0147]

[0148] where || denotes vector concatenation, ReLU is a rectified linear unit activation function, and W and b are weight parameters.

[0149] The message passing block is followed by a multi-layer perceptron (MLP) with ReLU activation, whose task is to learn a non-translational relationship of the node features.

[0150] A state transition function of a general GCA can be implemented with a GNN, which is denoted as a Graph Neural Cellular Automata (GNCA) in the present disclosure, which contains all the necessary elements to describe the GCA transition rules.

[0151] First, d is the dimension of the vector attribute function h i , and d' is the dimension of the GNN message passing scheme function h' i . When the GNN is configured as d = d', then both functions are homomorphisms that only depend on the node i and its neighbors . Then, by considering the state of the GCA can be interpreted as a graph signal, e.g., h i = s i .

[0152] Secondly, the isotropic or anisotropic conversion rule is modeled by a graph neural network.

[0153] Finally, the CA rule with a larger range of neighbors can be implemented by GNCA of the present application by stacking multiple message passing layers or redefining the neighbor function to include higher order neighbors.

[0154] The present application uses to represent the conversion function of GCA, and uses to represent the conversion function of GNCA, and λ is a learnable parameter, and d represents the dimension of the state. The dependence of any one function on the graph is implicit.

[0155] The equivalent calculation of the present application can be specifically described as implementing GCA with M state spaces, which is equivalent to implementing an arbitrary M-state CA with CNN, that is, the equivalent calculation of GNN and GNCA, which depends on two main neural computing blocks:

[0156] 1. Two convolutional layers with 1x1 filters to calculate the one-hot encoding of the state;

[0157] 2. A convolutional layer with 3x3 filters to match all possible input patterns of the conversion rule and calculate the corresponding output.

[0158] The general principle of designing CA conversion rules with CNN can be directly extended to GCA and GNN.

[0159] This process is divided into two stages:

[0160] Stage 1: The first neural computing block remains unchanged (i.e., equivalent).

[0161] Consider a GCA with a state space of and M cells, and the maximum node degree d max ≤ M (usually d max << M). Because the first neural computing block is a conversion applied to each node independently of its neighbor nodes.

[0162] Stage 2: The second neural computing block remains unchanged.

[0163] First, CNN covers the most general case where the conversion rule is anisotropic. By extending this principle to the case of GCA, the present application requires each cell to have up to d max neighbor nodes, which are identified by a unique one-hot vector edge attribute. The pattern matching mechanism can be used to shape d maxThe weight matrix of size N x N is implemented according to the properties of its associated edges, which is used to compute the message from each neighbor node. However, this case represents an upper bound of complexity, because many GCA rely on isotropic rules, and better implementations can be obtained if the structure of the GCA rules is known. For example, Conway's Game of Life can be implemented with a CNN, where one convolutional layer with 3 x 3 x 5 filters, 1 channel to implement the identity, 4 channels to compute the neighbor nodes, and 2 convolutional layers with 1 x 1 x 5 filters for pattern matching.

[0164] Finally, the 1 x 1 x 5 convolutional layer is equivalent to the fully connected layer described for the post-processing in GNNs. Extending to continuous state spaces, the expressive power of the architecture described for GNNs is enough to represent any transition function of a GCA with continuous state spaces.

[0165] From the general conclusions of machine learning, a two-layer MLP with a sufficient number of neurons is a universal approximator of any Borel measurable function. Therefore, the pre-processing MLP of the general GNCA architecture is enough to compute any state representation required. The GNCA architecture allows modeling any permutation-invariant function of the neighbor nodes. By concatenating the representation of a node with the representation of its neighbors and using the result as input to the post-processing MLP, the present invention is also able to compute any transition of the state.

[0166] Cellular automata produce complex patterns

[0167] In GCA, a similar anisotropic behavior can be obtained by uniquely enumerating the neighbors of each cell and adjusting the transition rules accordingly.

[0168] The most basic cellular automaton local transition rule can be seen as a Boolean function of the points in the neighborhood. The transition rule is allowed to treat the neighbors of a cell differently depending on the position of the cell.

[0169] According to this rule, the value of a particular position is the sum modulo of the values of its two neighboring positions at the previous time step.

[0170]

[0171] s t (d) is the value of the point d at time step t. represents the modulo addition operation.

[0172] According to the cellular automaton rule, The previous and next elements always have symmetry.

[0173] The Boolean function representation of the cellular automaton is convenient for implementation on a standard serial processing digital computer.

[0174] The cellular automaton rules have the important simplifying feature of additive superposition or additivity, according to which evolution satisfies the superposition principle:

[0175]

[0176] Step four, orbit finding: apply the multiple graph neural networks in step three to a new particle collision experiment. Input the multiple sets of measurements of the same particle in the new particle collision experiment into the multiple graph neural networks to obtain multiple different predicted orbits.

[0177] From the known physical properties of the particle, find the real orbit of the particle from the different predicted orbits and infer the new physical properties of the particle. Physicists compare the results output by the multiple graph neural networks and select the set of measurements that best fit the known properties of the particle as the particle's orbit.

[0178] In this step, orbit finding takes a set of measurements in a candidate orbit and estimates as accurately as possible a set of parameters describing the state of the particle at some place in the detector, usually at a reference surface close to the particle beam.

[0179] Orbit finding takes a set of measurements in a candidate orbit as a starting point. Its goal is to estimate as accurately as possible a set of parameters describing the state of the particle at some place in the detector, usually at a reference surface close to the particle beam.

[0180] Candidate orbits refer to the possible orbits formed by the questionable hit points in the orbit finding process. These hit points are usually disturbed by external noise, and the noise formed hit points will change the determination of the orbit. For these questionable hit points, the possible orbits formed by them are retained as candidate orbits.

Claims

1. A particle orbit discovery method based on graph neural network cellular automata, characterized in that, The method comprises the following steps: Step one, data generation: using a high-energy physics detector to record a plurality of particle hit data generated in a particle collision event, including the position, direction and momentum of the particle at its hit point; Step two, training a graph neural network: using a plurality of particle hit data generated in a plurality of particle collision events to supervise the training of a graph neural network; Step three, introducing a graph cellular automaton: the graph neural network in step two is equivalent to a graph neural cellular automaton; the conversion rules of the graph neural cellular automaton are added to a conversion rule library; a plurality of graph neural cellular automaton conversion rules are selected from the conversion rule library to equivalent calculate a plurality of corresponding graph neural networks; Step four, orbit discovery: the plurality of graph neural networks in step three are applied to a new particle collision experiment; a plurality of different predicted orbits are obtained by inputting a plurality of measurement values of the same particle obtained in the new particle collision experiment into the plurality of graph neural networks; and the real orbit of the particle is discovered from the different predicted orbits according to the known physical properties of the particle.

2. The particle orbit discovery method based on graph neural network cellular automata according to claim 1, characterized in that, The step one comprises: First stage: a group of high-energy physics detectors are placed near the particle collision area; Second stage: the charged particles generated in the collision will ionize the sensitive layer of the material in the collision accelerator tube when leaving the collision area, and will generate a plurality of particle position measurement values along the orbit of each particle; the high-energy physics detector records the position of all particles at certain time intervals, and all particles measured at the same time will form a circle, which is called a detector layer; periodic measurement forms a plurality of detector layers, and all the detector layers form a plurality of concentric circles; the detector layers measured at adjacent time intervals are defined as adjacent layers.

3. The particle orbit discovery method based on graph neural network cellular automata according to claim 2, characterized in that, In the first stage, the high-energy physics detector is immersed in a strong magnetic field of a curved orbit to minimize the amount of unrelated substances in the high-energy physics detector; and the measurement of the momentum component uses the principle that the curvature is proportional to the momentum component transverse to the magnetic field.

4. The particle orbit discovery method based on graph neural network cellular automata according to claim 2, characterized in that, In the step two, the plurality of particle hit data generated in the particle collision event is represented by a graph structure connected by hit points; if the hit points meet the graph neural network judgment rules, they will be connected with the hit points of the adjacent layers; the hit point refers to the measurement record of the position of the current particle by the high-energy physics detector at certain time intervals; the particle hit data refers to the measurement record of the position of all particles by the high-energy physics detector at the same time; in the graph neural network, the graph node corresponds to the hit point, and the edge corresponds to the connection of the hit point.

5. The particle orbit discovery method based on graph neural network cellular automata according to claim 4, characterized in that, In the step two, the graph neural network is composed of an input conversion layer and a plurality of cyclically alternating edge networks and node networks; the final output layer is an edge network; the input of the graph neural network is node features and connectivity rules; the node features refer to hit coordinates in three-dimensional space; the graph neural network performs two different tasks: 1) a binary hit classification model, which learns to identify a local orbit in the graph by classifying graph nodes; 2) a segment classifier model, which learns to identify multiple orbits at a time by classifying edges in the graph; The edge network uses the features of the start node and the end node to calculate the weight of each edge of the graph; the node network uses the nodes on the previous detector layer and the next detector layer respectively, inputs the current features of the nodes and the edge weight values between the nodes into an aggregation function, and thus calculates the new features of each node.

6. The particle orbit discovery method based on graph neural network cellular automata according to claim 5, characterized in that, Both the edge network and the node network are implemented by multi-layer perceptrons, each of which has two hidden layers and a hyperbolic tangent activation function.

7. The graph neural network based cellular automata method for particle orbit discovery according to any one of claims 1 to 6, characterized in that, The step three, the anisotropy in the cellular automaton is regarded as a dependency relationship of the conversion rule, the dependency relationship represents a specific relationship existing between the cell and the neighbor, which refers to taking the union of the current state of node i and the current state of all neighbor nodes j of node i, and taking the union of the edge weight between node i and all neighbor nodes j of node i as the next state of node i, that is, s i ′ , so that different neighbor nodes j of i can be treated differently according to the edge weight of node j relative to node i.

8. The particle orbit discovery method based on graph neural network cellular automata according to claim 7, characterized in that, The state transition function of the graph cellular automaton is implemented by using the graph neural network, and a graph neural cellular automaton is obtained, wherein: First, when the graph neural network is configured as d = d ′ , the state of the graph cellular automaton is interpreted as a graph signal; d is the dimension of the vector attribute function h i , d ′ is the dimension of the graph neural network message passing scheme function h i ′ ; Secondly, the isotropic or anisotropic transition rule is modeled by using the graph neural network; Finally, the graph neural cellular automaton is stacked with multiple message passing layers or the neighbor function is redefined to contain higher-order neighbors, so as to realize the cellular automaton rule with a larger range of neighbors.

9. The particle orbit discovery method based on graph neural network cellular automata according to claim 8, characterized in that, The graph neural network and the graph neural cellular automaton are equivalent in calculation by using M state spaces to implement the graph cellular automaton, which depends on two neural computing blocks: The first neural computing block is two convolutional layers with a 1x1 filter, which is used to calculate the one-hot encoding of the state; The second neural computing block is a convolutional layer with a 3x3 filter, which is used to match all possible input patterns of the transition rule and calculate the corresponding output; The transition rule design process is divided into two stages: The first stage: the first neural computing block remains unchanged; Consider a state space as A graph cellular automaton with M units has a maximum node degree of d. max ≤M, the first neural computation block is a transformation applied individually to each node, independent of its neighboring nodes; The second stage: the second neural computing block remains unchanged; Each cell has d max neighbor nodes, which are identified by a unique one-hot vector edge attribute, the pattern matching mechanism is implemented with a weight matrix of shape max N x N, which is used to compute the message from each neighbor node according to its associated edge attribute; extended to continuous state space, the expressiveness of the architecture described by the graph neural network is sufficient to represent any transition function of a graph cellular automaton with continuous state space.

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