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

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.

Traditional neural networks used to model physical systems often ignore fundamental conservation laws—like mass or momentum conservation—so they can drift into predictions that are physically impossible.We introduce Riemann Tensor Neural Networks (RTNNs), a new architecture that embeds these conservation principles directly into its design by outputting specially structured tensors that are divergence-free by construction. This means RTNNs fields never violate mass or momentum conservation, down to machine precision. We also prove that RTNNs can approximate any smooth conserved flow as accurately as desired.By hard-wiring physics into the model, RTNNs produce more physically consistent predictions on standard fluid-flow benchmarks compared to previous approaches. This makes them powerful surrogate models for complex simulations that can be written in conservation form.