derive conservation law

This work addresses the challenge of automatically discovering conservation laws from noisy trajectory data. Methodologically, it introduces a hybrid framework that decouples learning and symbolic search: a neural ODE models continuous dynamics, while a Transformer generates symbolic candidate invariants; a symbolic-numerical hybrid verification mechanism enables multi-stage filtering and refinement. The key contribution is the first end-to-end joint optimization of dynamical system modeling and symbolic invariant generation, significantly enhancing robustness to noise and discovery accuracy. Experiments on canonical physical systems demonstrate that the method reliably recovers known conservation laws—even under low signal-to-noise ratios (as low as 5 dB)—and identifies several novel, physically meaningful candidate invariants. It consistently outperforms baseline approaches that directly fit trajectories, achieving superior performance across all evaluation metrics.

This study addresses the unclear relationship and applicable conditions between parameter symmetries and conservation laws in gradient flow. By integrating differential geometry with an analogy to Noether’s theorem, it constructs a unified geometric framework to elucidate their intrinsic connection. The work introduces the novel concept of “combinatorial identifiability” and establishes a general inheritance principle for conservation laws in multi-layer networks. Furthermore, it successfully characterizes the symmetry and conserved quantity structures in multi-head attention, polynomial, and deep linear networks. Overall, this research provides a systematic geometric perspective for the theoretical analysis of deep learning architectures.

This work addresses the challenge of discovering conservation laws from data in the presence of parameter variations, non-polynomial forms, local minima, and spurious correlations—particularly in chaotic systems. The authors propose NGCG, a neurosymbolic pipeline that decouples dynamics learning from invariant discovery: it first learns approximately constant latent representations by minimizing variance across multiple initializations, then employs diverse symbolic regression techniques to generate candidate expressions. Rigorous constancy gating and diversity-based filtering are introduced to eliminate false positives. NGCG achieves zero false discoveries for the first time, attaining perfect scores (DR = 1.0, FDR = 0.0, F1 = 1.0) across nine benchmark systems—including chaotic, dissipative, and partial differential equation models—with conservation-law constancy errors two to three orders of magnitude lower than the best baseline. The method also demonstrates robustness to noise, high sample efficiency, hyperparameter insensitivity, and runs in minutes per system.

This work addresses the challenge of simultaneously preserving conservation, entropy stability, and hyperbolicity in data-driven learning of hyperbolic conservation laws. To this end, the authors propose the SymCLaw framework, which parameterizes the flux function and jointly learns a convex entropy function along with its associated entropy potential. Without requiring prior knowledge of the governing equations, SymCLaw is the first data-driven approach to unify these three fundamental physical properties and automatically select physically admissible weak solutions. By integrating entropy-stable numerical fluxes with standard discretization-compatible techniques, the method demonstrates strong generalization to unseen initial conditions, robustness to noise, and high-accuracy long-time predictions across benchmark problems including Burgers’, shallow water, Euler, and KPP equations.

This work systematically identifies and rectifies seven critical errors in Liu, Madhavan, and Tegmark’s machine learning–based discovery of conservation laws from a one-dimensional damped harmonic oscillator—including mis-specified physical priors, conflation of mathematical definitions of conserved quantities, inappropriate error metrics, and omission of differential equation validation. To address these, we introduce a physics-constrained error analysis framework, rigorous analytical verification, and formal scrutiny of conservation law definitions, thereby falsifying the original method’s applicability to dissipative systems. Our principal contributions are threefold: (i) establishing the first empirically testable theoretical validation standard for ML-driven conservation law discovery; (ii) rigorously distinguishing *invariance* (under symmetry transformations) from *conservation* (time-independence along trajectories); and (iii) proposing a robust modeling paradigm integrating differential geometry and dynamical systems theory—substantially enhancing methodological rigor and reproducibility in physics-guided machine learning.

This work addresses the limited understanding of conservation laws governing gradient flow in modern neural networks, which hinders explanations of the implicit bias in over-parameterized models. We systematically extend conservation law theory to mainstream architectures, including feedforward networks with GELU, SiLU, or SwiGLU activations; multi-head attention mechanisms incorporating sinusoidal and rotary positional encodings; and mixture-of-experts models featuring diverse gating designs. By integrating gradient flow analysis, invariant theory from differential equations, and deep learning architecture modeling, we derive key conserved quantities for each architecture and empirically validate their existence and stability. Our findings uncover invariant structures within gradient dynamics, offering a novel perspective for understanding the inductive biases inherent in contemporary neural networks.

This study investigates the training dynamics of coupled learning (CL) and equilibrium propagation (EP) in the continuous-time, small-perturbation limit, revealing a parameter conservation law in physically realizable systems that parallels mass conservation. By leveraging continuous-time dynamical systems analysis and perturbation theory, the work establishes— for the first time—that this conservation law holds universally across a broad range of physical settings. Furthermore, it elucidates how this constraint governs the convergence behavior of learning in linear circuits. The findings not only enhance the reliability of CL and EP training but also provide a rigorous theoretical foundation and practical guidance for achieving efficient and stable learning in neuromorphic hardware implementations.

本文从拉格朗日视角出发，通过分析连续去噪器的局部泰勒展开，推导出一种新的流匹配方法，并利用特征线法解决了轨迹曲率问题。

This work addresses the tendency of unconstrained neural ordinary differential equations (Neural ODEs) to violate domain-specific invariants—such as physical conservation laws—in scientific simulations, leading to distorted long-term predictions. To resolve this, the authors propose an invariance compiler framework that, for the first time, treats scientific invariants as first-class constructs in Neural ODE architecture design. Leveraging large language model–driven program synthesis, the framework automatically transforms generic Neural ODE specifications into structure-preserving models whose trajectories remain strictly confined to valid manifolds. By construction, the resulting models guarantee physical consistency—subject only to numerical integration error—without requiring post-hoc regularization, and they enforce prescribed invariants exactly in continuous time. This approach significantly enhances the credibility and physical plausibility of long-term simulations and establishes a systematic, cross-domain design paradigm for invariant-aware scientific machine learning.

This work investigates whether intrinsic symmetries in training data induce conserved quantities during gradient flow training of neural networks. By integrating tools from differential geometry and dynamical systems theory, the study establishes—for the first time—a systematic connection between data symmetries and conservation laws in training dynamics, employing tensorized networks (including linear, polynomial, and Lightning Attention architectures) as an analytical framework. The analysis demonstrates that, under general non-polynomial losses, data symmetries do not yield additional conserved quantities; however, when combined with data augmentation under mean squared error (MSE) loss, novel conserved quantities emerge. This finding uncovers a distinctive conservation mechanism specific to MSE loss and offers a new perspective for understanding the dynamics of neural network training.