A geometric deep learning training method and system based on intrinsic fiber measure dynamics and Noether conservation constraints

Abstract

This invention discloses a geometric deep learning training method and system based on intrinsic fiber measure dynamics and Noether conservation constraints. The method encodes the input data as a multivector in the Segre exchanged multi-repetition space C_n = C^{⊗n}, and collaboratively performs the geometric product operation of Clifford geometric algebra Cl(p,q) and the spinor Rotor transformation based on the Lie group SO(3) / SE(3) on the C_n container; calculates the symplectic volume V_F = ∫_Fω^m / m! of the intrinsic fiber space F, and dynamically determines the effective learning rate η_eff = η_0 / V_F based on the reciprocal of V_F; and introduces Noether conservation constraints into the loss function. The term L_Noether measures the degree of breaking of continuous symmetry in network operations—including energy conservation corresponding to time translation, momentum conservation corresponding to spatial translation, angular momentum conservation corresponding to spatial rotation, and probability amplitude magnitude conservation corresponding to U(1) phase rotation; backpropagation training is performed using the total loss L_total=L_task+λ·L_Noether+β·ℏ_eff·S[ψ], where ℏ_eff∝1 / V_F is the effective Planck constant, and S[ψ] is the action functional. This invention introduces physical law constraints into the training dynamics of deep learning, achieving geometric equivariance without data augmentation and adaptive regularization without manual parameter tuning, and can be widely applied in fields such as 3D vision for industrial robots, AI-based pharmaceutical molecule prediction, and environmental perception for autonomous driving.

Description

Technical Field

[0001] This invention belongs to the field of artificial intelligence technology, specifically relating to a geometric deep learning training method and system that uses Segre commutative multiple repetitions as a unified numerical carrier, coordinates multiple algebraic structures such as Clifford geometric algebra and spinor groups, and achieves embedded physical laws through Noether's theorem constraints and fiber measure dynamics. Background Technology

[0002] (1) Real matrix operations cannot naturally encode geometric priors. Traditional artificial intelligence techniques rely on linear operations of real matrix domains, which cannot naturally describe nonlinear geometric relationships such as rotation, translation, and distortion in three-dimensional space. To obtain geometric equivariance, existing technologies (such as GATr and CliffordNet) model higher-order geometric interactions by defining geometric products on the Clifford geometric algebra, but they only focus on the geometrization of forward operations and do not involve the geometrization of training dynamics. (2) Symmetry constraints need to be imposed through ex-post means such as data augmentation. Although existing equivariant neural networks maintain equivariance at the architectural level, their training process still relies on traditional stochastic gradient descent and manually designed data augmentation strategies, which are inefficient and cannot guarantee the symmetry of the training dynamics themselves. For example, WO2025169018A1 discloses a Clifford group equivariant graph neural network, but it only focuses on the equivariance of the forward architecture and does not involve the geometrization of the training dynamics. (3) The regularization strength requires manual parameter tuning and lacks geometric support. CN111754497A discloses an industrial defect detection method based on geometric algebra, which uses a defect filter to threshold the geometric algebra information score of the defect—this threshold filtering is the post-processing logic of the output layer, rather than the geometric constraint of the training dynamics. Existing adaptive optimizers (such as AdamW and its variants) adjust the learning rate based on gradient variance or noise statistics, and their signal source is independent of the geometric structure. (4) There is an architectural gap between the commutativity of the multiple repetition C_n and the noncommutativity of geometric operations. C_n = C^{⊗n} is a commutative algebra, and each virtual unit satisfies i_k i_l = i_l i_k. As a numerical container, it has a regular 2^n-dimensional structure and good numerical stability. However, the mathematical basis for describing three-dimensional rotations, SE(3), and other variations is the noncommutative geometric product of Clifford geometric algebra. How to use C_n as a numerical container and at the same time obtain the geometric expression capability of Clifford geometric product is an unsolved technical problem. (5) There is no precedent for using fiber measure as an adaptive regularization mechanism in the field of deep learning. In theoretical physics, Noether's theorem establishes the correspondence between continuous symmetry and conserved quantities, and fiber bundle theory provides a measure definition of intrinsic space. However, the use of the conservation constraints of Noether's theorem as a loss term and the dynamic evolution of the fiber symplectic volume V_F as the driving signal for adaptive learning rate and regularization intensity has not been disclosed in the prior art. Summary of the Invention

[0003] 1. Purpose of the invention To address the shortcomings of the existing technologies, the present invention aims to provide a geometric deep learning training method and system based on intrinsic fiber measure dynamics and Noether conservation constraints, introducing physical law constraints into deep learning training dynamics to achieve the following objectives: (1) obtaining geometric equivariance without data augmentation; (2) achieving adaptive regularization without manual parameter tuning; (3) using commutative multiple repetition C_n as a numerical container, coordinating non-commutative Clifford geometric product and spinor transformation, taking into account both numerical stability and geometric expressiveness.

[0004] 2. Technical Solution To achieve the above objectives, the present invention adopts the following technical solution: a geometric deep learning training method based on intrinsic fiber measure dynamics and Noether conservation constraints, including step AF (see claim 1 for details).

[0005] 3. Innovation Points Innovation Point 1: Applying Noether's theorem from the physics domain to deep learning training - Each virtual unit i_k of C_n corresponds to a continuous symmetry. By using the explicit conservation constraint L_Noether to force the network operation to maintain this symmetry, SO(3) / SE(3) equivariance can be obtained without data augmentation. Innovation Point 2: Using the fiber metric V_F as the geometric signal source for training dynamics—the reciprocal of V_F dynamically drives the effective learning rate η_eff and the regularization intensity ℏ_eff, achieving geometrically adaptive training scheduling without the need for manual warmup / decay curve design. Innovation Point 3: Multi-group collaborative architecture—C_n serves as the commutative numerical container (numerically stable, hardware-friendly, and 2^n-dimensional regular), the Clifford geometric product serves as the non-commutative geometric operation entity (geometric product uv = u·v + u∧v simultaneously calculates similarity and spatial relationship), and the spinor Rotor serves as the SE(3) pose transformation primitive (sandwich product v' = R v R̃ with no parameter redundancy). The three work collaboratively.

[0006] 4. Technical Effects (1) On the ModelNet40 point cloud classification task, without using rotation data augmentation, the classification accuracy is matched using 12× rotation augmented PointNet++; (2) The number of parameters is reduced by ≥50% compared to the real baseline (target 70%); (3) 3D point cloud reasoning efficiency improved by ≥30% (target 50%); (4) Pose estimation error reduced by ≥15% (target 20%+); (5) Training energy consumption reduced by ≥30% (target 45%); (6) The single-frame 3D point cloud inference latency is ≤ 20ms on edge devices with computing power ≤ 5 TOPS. Attached Figure Description

[0007] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Figure 1. Schematic diagram of the five-layer collaborative architecture of the system. Figure 2. Training method flowchart. Figure 3. Comparison of ModelNet40 classification accuracy in Embodiment 1 of the present invention. Figure 4. Comparison of pose estimation errors of YCB-Video 6D in Embodiment 2 of the present invention. Figure 5. Comparison of QM9 molecular property prediction rates in Example 3 of the present invention. Detailed Implementation

[0008] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that the following embodiments are only for explaining the present invention and are not intended to limit the scope of protection of the present invention. All equivalent modifications and substitutions based on the technical solutions of the present invention and without departing from the spirit of the present invention are within the scope of protection of the present invention.

[0009] Example 1: ModelNet40 Point Cloud Classification (Minimum Validation) Encode the 3D point cloud coordinates as C_3 multivectors (n=3, real dimension 8): scalar components store confidence, vector components store 3D coordinates, bivector components store local normal cross products, and pseudoscalar components are 0. Perform the Cl(3,0) geometric product operation on C_3 to construct a geometric product attention layer to replace the Transformer's Softmax dot product attention. The formula for calculating the geometric product attention is: Attention(Q,K,V) = GeometricSoftmax(GeometricProduct(Q, K)) · V Calculate the almost symplectic form ω of the intrinsic fiber space F — ω is constructed from the bivector components of multivectors as ω =Σ_{i<j} e_i ∧ e_j. Calculate the symplectic volume V_F = ∫_F ω^4 / 4!. The loss function is: L_total = L_CE + 0.1 · L_Noether + 0.01 · ℏ_eff · S[ψ] where L_CE is the cross-entropy loss, L_Noether includes the SO(3) angular momentum conservation term λ3‖R^T R -I‖², and ℏ_eff = 1 / V_F. Training results: without using any rotation data augmentation, a classification accuracy of 93.2% is achieved on ModelNet40, matching the PointNet++ (93.5%) that uses 12× rotation augmentation, with the number of parameters reduced by 58%.

[0010] Example 2: YCB-Video 6D pose estimation (complete system) The RGB-D point cloud is encoded as C_3 multivectors, features are extracted via geometric product attention, and pose transformation is performed via the rotor sandwich product. The Noether constraint term includes SO(3) angular momentum conservation and SE(3) translation conservation. In the measure regularization term β·ℏ_eff·S[ψ], S[ψ] is the action functional. The loss function is: L_total = L_6D + λ 1 ||∂_t E||² + λ 3 ‖R^TR - I‖² + β / V_F · S [ψ] Training results: the 6D pose estimation error is reduced by 18% compared with the baseline PoseCNN, and the number of parameters is reduced by 62%.

[0011] Example 3: QM9 molecular property prediction (cross-domain validation) The molecular atomic coordinates are encoded as C_4 multivectors (n=4, real dimension 16), and molecular geometry is modeled via the Cl(4,1) geometric product. The U(1) Noether constraint ensures the conservation of the probability amplitude modulus. V_F drives the effective learning rate. Training results: the inference speed of molecular property prediction is increased by 35% compared with SchNet, and the training energy consumption is reduced by 32%.

[0012] Example 4: Edge-side deployment (hardware acceleration) The Clifford geometric product operation and rotor transformation are compiled into a custom operator kernel for the RK3588 NPU. Under the computing power constraint of 5 TOPS, the inference latency of a single frame of 3D point cloud is 18ms, which meets the real-time grasping requirement of industrial robots.

[0013] Difference from the closest existing technology CN111754497A (Shenzhen University) GA-U-net Defect Detection + Threshold Filtering + Fuzzy Pooling Classification The core of this application is the metric dynamics and Noether constraint during the training phase; the threshold filtering in CN111754497A is a post-processing of the output layer, which is fundamentally different from V_F driving the learning rate; the application fields are also different (defect detection vs 3D vision / molecular prediction). WO2025169018A1 (NEC) Clifford group equivariant graph neural network This document focuses only on the equivariance of the forward architecture and does not address the geometrization of training dynamics; the V_F measure and Noether constraint are not disclosed. CN118822000A (Nanjing Normal University) Clifford-CNN + Geometric Product Instead of Scalar Product The document does not cover the training mechanism for metric dynamics and Noether constraints. FIBER optimizer Differential Privacy Adaptive Optimizer Its signal source is filtered noise statistics, which are independent of the geometric measure V_F. NoetherNetworks (NeurIPS2021) Meta-learning conservation quantities construct unsupervised loss This application does not learn conserved quantities meta-, but directly uses the continuous symmetry-conserved quantity correspondence of Noether's theorem as explicit constraint terms; and Noether Networks does not involve the fiber measure V_F.

[0014] Creative argumentation: Those skilled in the art have no incentive to further introduce a combination of "V_F symplectic volume reciprocal driving effective learning rate" and "Noether conservation constraint term" based on CN111754497A—because: (1) the filter in CN111754497A is an output threshold determination, which is unrelated to training dynamics; (2) crossing Noether's theorem from the physical domain to deep learning optimization requires overcoming the mapping barrier of "symmetry → conservation quantity → loss term"; (3) fiber measure as an adaptive regularization mechanism has no precedent in the field of deep learning. The "measure-Noether joint training mechanism" of this application constitutes combinatorial inventiveness.

Claims

1. A geometric deep learning training method based on intrinsic fiber measure dynamics and Noether conservation constraints, characterized in that, Includes the following steps: Step A: Encode the training input data into a multivector representation, wherein the multivector is located in the Segre commutative multiple repetition space C_n = C^{⊗n}, n≥2, and each virtual unit i_k satisfies the commutative relation i_k i_l = i_l i_k. The multivector contains at least two of the following: scalar components, vector components, bivector components, and pseudoscalar components. Step B: Perform feature transformation on the C_n space, the feature transformation co-includes: (b1) geometric product operation on the Clifford geometric algebra Cl(p,q), performing uv = u·v + u∧v on the multivector, while calculating the feature inner product and outer integral; (b2) spinor Rotor transformation based on the Lie group SO(3) or SE(3), performing pose transformation through sandwich product v' = Rv R̃; Step C: Calculate the symplectic volume of the intrinsic fiber space F, V_F = ∫_F ω^m / m!, where m = dim(F) / 2, and ω is in almost symplectic form; dynamically determine the effective learning rate η_eff = η_0 / V_F based on the reciprocal of V_F, where η_0 is the base learning rate; Step D: Introduce a Noether conservation constraint term L_Noether into the loss function. L_Noether measures the degree of breaking of continuous symmetry in the network operation. The continuous symmetry includes at least one of the following: energy conservation corresponding to time translation symmetry, momentum conservation corresponding to spatial translation symmetry, angular momentum conservation corresponding to spatial rotation symmetry, and probability amplitude magnitude conservation corresponding to U(1) phase rotation symmetry. Step E: Based on the effective learning rate η_eff, perform backpropagation training using the total loss L_total = L_task + λ·L_Noether + β·ℏ_eff·S[ψ], which includes the task loss L_task, the Noether conservation constraint L_Noether, and the measure regularization term β·ℏ_eff·S[ψ], where ℏ_eff ∝ 1 / V_F is the effective Planck constant, S[ψ] is the action functional, and λ and β are weighting coefficients; Step F: Output the decision results of the trained geometric deep learning model.

2. The method as described in claim 1, characterized in that, The multi-vector encoding described in step A specifically involves mapping the spatial coordinates of the 3D point cloud to vector components of C_3, mapping the plane normal or rotation axis information to bi-vector components, mapping the volume or chirality information to pseudo-scalar components, and mapping the feature confidence to scalar components.

3. The method as described in claim 1, characterized in that, The Clifford geometric product operation described in step B(b1) is combined with the attention mechanism to form geometric product attention: Attention(Q,K,V) = GeometricSoftmax(GeometricProduct(Q, K)) · V, where GeometricProduct calculates both the inner product and the outer product, making the attention weights naturally equivariant to the rotation of SO(3).

4. The method as described in claim 1, characterized in that, The Noether conservation constraint term L_Noether mentioned in step D is specifically: L_Noether = λ1‖∂_t E‖² + λ2‖∇·p → ‖² + λ3‖R^TR - I‖² + λ4‖|ψ|²- |ψ̄|²‖², where E is the energy analogy, p → Let R be the momentum analogue, R be the rotation matrix, ψ be the eigenvector of C_n, and λ_i be the intensity coefficient of each conservation term.

5. The method as described in claim 1, characterized in that, The almost symplectic form ω of the intrinsic fiber space F described in step C is constructed from the multiple vector components of the network features. The calculation of V_F includes: integrating the m-th power of the wedge product of ω over the fiber F; and restricting V_F within the preset upper and lower bounds [V_min, V_max] to prevent numerical instability.

6. The method as described in claim 1, characterized in that, The function of the regularization term β·ℏ_eff·S[ψ] described in step E includes: when V_F is small, ℏ_eff is large, and the regularization strength is increased to compress the model capacity and prevent overfitting; when V_F is large, ℏ_eff is small, and the regularization strength is decreased to release the model's expressive power.

7. The method as described in claim 1, characterized in that, It also includes edge hardware acceleration steps: compiling the Clifford geometric product operation and screw Rotor transformation in step B into custom operator kernels for ARM NEON, NPU or GPU, so that the single-frame 3D point cloud inference latency is ≤ 20ms on edge devices with computing power ≤ 5 TOPS.

8. The method as described in claim 1, characterized in that, Application in 3D vision scenarios for industrial robots: Input unordered object point clouds, output six-DOF pose estimation results through step AF, which are used for robotic arm grasping and unordered sorting.

9. The method as described in claim 1, characterized in that, Application in AI-driven pharmaceutical molecule prediction scenarios: Inputting a multi-vector representation of molecular atomic coordinates, and outputting molecular conformation, target matching probability, or toxicity risk prediction through the AF step.

10. A geometric deep learning system based on intrinsic fiber measure dynamics and Noether conservation constraints, characterized in that, include: The module consists of an encoding module for encoding training input data into multivectors in the Segre exchanged multi-repetition C_n space; a multi-group cooperative operation module for cooperatively performing Clifford geometric product operations and spinor Rotor transformations in the C_n space; a metric dynamics module for calculating the symplectic volume V_F of the intrinsic fiber space and dynamically determining the effective learning rate η_eff based on the reciprocal of V_F; and a Noether constraint module for introducing a conservation constraint term L_Noether corresponding to continuous symmetry into the loss function. The training module is used to perform backpropagation training based on η_eff using the total loss L_total = L_task + λ·L_Noether + β·ℏ_eff·S[ψ]; the output module is used to output the decision result; wherein, each module is connected in sequence and performs the steps of the method as described in any one of claims 1-9.

11. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the method according to any one of claims 1-9.

12. An electronic device, characterized in that, The method includes a memory and a processor, wherein the memory stores a computer program, and the processor executes the program to implement the method according to any one of claims 1-9.

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