Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks
This work provides a novel physics-constrained neural architecture for continuum mechanics, addressing conservation law enforcement in PDE surrogates, though it is incremental in applying known tensor constraints to neural networks.
The paper tackled the problem of learning conservative systems by introducing Riemann Tensor Neural Networks (RTNNs), which inherently satisfy divergence-free symmetric tensor conditions to enforce conservation laws like mass and momentum, achieving improved accuracy in PDE benchmarks.
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.