conservation-law analysis

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.

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 work addresses the problem of automatically discovering symmetries and corresponding conservation laws (e.g., energy, linear momentum, angular momentum) of physical systems directly from noisy, discrete trajectory data—without requiring velocity/momentum measurements or prior knowledge of the governing model structure. Methodologically, it introduces the first end-to-end framework for jointly learning discrete Lagrangian functions and their symmetry groups, integrating discrete variational integration, Lie group theory, and neural network parameterization, alongside a symmetry-driven loss function. It further proposes variational backward error analysis to rigorously unify discrete modeling with continuous physical constraints. Experiments demonstrate that the method significantly improves long-term trajectory prediction accuracy under noise, strictly preserves conservation quantities, and achieves superior qualitative and quantitative performance compared to state-of-the-art baselines.

This paper addresses the unified modeling of symmetries in machine learning. It proposes a framework grounded in differential geometry and convex optimization to (1) enforce known symmetries, (2) automatically discover unknown symmetries in models or data, and (3) actively induce symmetry breaking via user-specified candidate groups. The core contribution is the first formulation of symmetry imposition and discovery as dual linear-algebraic tasks, leveraging the Lie derivative to characterize fiberwise linear Lie group actions on vector bundles, and employing nuclear-norm relaxation to construct convex regularization terms. The method is broadly applicable to neural networks, dynamical system discovery, basis-function regression, and neural operators. Empirically, it significantly improves generalization performance and parameter efficiency—particularly in low-data regimes—while preserving geometric structure and interpretability.

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 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 investigates how to automatically uncover low-dimensional constraint structures induced by symmetries and conservation laws from high-dimensional physical data in the absence of explicit prior knowledge. To this end, we propose an unsupervised representation learning framework based on variational autoencoders that eschews conventional designs relying on explicit symmetry embeddings. Instead, our approach leverages the information bottleneck principle to drive the latent space to self-organize and reveal the dimensionality reduction inherent to underlying symmetries. Evaluated on geometric systems and particle physics datasets, the method successfully recovers theoretically expected symmetry structures and systematically delineates the theoretical limits and practical challenges of symmetry inference under minimal inductive bias.

While existing linear probes can decode conserved quantities with high accuracy, they cannot determine whether a dynamics model causally relies on them for predictions. This work introduces the concept of “causal deployment,” combining single-step activation swapping, a causal effect metric (transfer-corr τ), and algebraic predicate analysis to systematically evaluate the causal role of conserved quantities in prediction. The study reveals that whether a model deploys conserved quantities depends on its training objective and is governed by the algebraic structure of its outputs; notably, even when probe R² ≈ 1, conserved quantities may remain causally inert. For the first time, the research demonstrates a strong correlation (r = +0.97) between deployment gaps and out-of-distribution (OOD) performance, validating this finding across diverse physical systems and a 158-million-parameter PDE foundation model.

Traditional structural causal models rely on directed acyclic graphs (DAGs), which struggle to represent symmetric constraints and dynamic systems with feedback loops. This work proposes causal zeros and causal differential equations, introducing activation operators to construct an extended causal framework that incorporates equilibrium manifolds and relative interventions with respect to attractors, thereby transcending the limitations of DAGs. Building on this foundation, the paper establishes an extended do-calculus, identifiability conditions, and counterfactual semantics tailored to such systems by integrating local solvability, graph admissibility, and temporal unfolding techniques. This theoretical advancement provides a unified foundation for causal reasoning in complex dynamical systems.

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.