用新神经网络自动满足质量动量守恒，提升物理模型精度
Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks
- 设计可精确满足守恒律的神经网络架构
- 在多个基准上实现更优预测准确率
- 适合需要严格守恒特性的物理仿真任务
散度为零的对称张量（DFST）是连续介质力学中的基础概念，用于描述质量与动量守恒等物理规律。本文提出黎曼张量神经网络（RTNN），一种能以机器精度自然满足DFST条件的新神经架构，为强制守恒律提供了强归纳偏置。我们证明了RTNN可任意精度逼近任意足够光滑的DFST，并验证其作为保守型偏微分方程代理模型的有效性，在多个基准测试中均取得更高精度。该工作首次将DFST作为归纳偏置引入神经型PDE代理模型，并在物理约束神经架构中显式实现了质量和动量的守恒。
原文摘要 · Abstract (English)
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.
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