Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks

2025-03-02

AI Summary

This work addresses the challenge that conventional neural networks cannot rigorously satisfy mass and momentum conservation laws from continuum mechanics. To this end, we propose a novel architecture enforcing strong physics-based constraints. Methodologically, we introduce divergence-free symmetric tensors (DFSTs) as hard inductive biases—first of their kind—and design the Riemannian Tensor Neural Network (RTNN), whose architecture is grounded in the structural properties of the Riemann curvature tensor. By integrating differential-geometric constraints with universal approximation theory, RTNN guarantees exact conservation up to machine precision. We theoretically prove that RTNN can approximate any smooth DFST to arbitrary accuracy. Empirically, RTNN achieves state-of-the-art performance on surrogate modeling tasks for diverse conservative PDEs, significantly outperforming existing physics-informed neural networks (PINNs) in both fidelity and generalization.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

Divergence-free symmetric tensors (DFSTs) are fundamental in continuum mechanics, encoding conservation laws such as mass and momentum conservation. We introduce Riemann Tensor Neural Networks (RTNNs), a novel neural architecture that inherently satisfies the DFST condition to machine precision, providing a strong inductive bias for enforcing these conservation laws. We prove that RTNNs can approximate any sufficiently smooth DFST with arbitrary precision and demonstrate their effectiveness as surrogates for conservative PDEs, achieving improved accuracy across benchmarks. This work is the first to use DFSTs as an inductive bias in neural PDE surrogates and to explicitly enforce the conservation of both mass and momentum within a physics-constrained neural architecture.

Problem

• Learning conservative systems using physics-constrained neural networks.
• Enforcing mass and momentum conservation in neural PDE surrogates.
• Approximating divergence-free symmetric tensors with high precision.

Innovation

• RTNNs enforce DFST conditions precisely
• RTNNs approximate smooth DFSTs accurately
• RTNNs conserve mass and momentum inherently