Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

arXiv:2505.19036 · math.NA, cs.NA, stat.ML · Submitted 2026-08-17 · Read on arXiv

Hanfei Zhou, Lei Shi

math.NA, cs.NA, stat.ML

Submitted: 2026-08-17

Updated: 2026-08-18

Code: https://github.com/hanfei27/wPINNonmaifolds

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Conservation Laws — 23× in this paper · explained in 3 episodes
• Activation Function — 11× in this paper · explained in 1 episode
• Error Decomposition — 8× in this paper · explained in 3 episodes
• Curse of Dimensionality — 6× in this paper · explained in 6 episodes
• Generalization — 6× in this paper · explained in 13 episodes
• Accuracy — 5× in this paper · explained in 1 episode
• Intrinsic Dimension — 4× in this paper · explained in 2 episodes
• Neural Network Approximation — 3× in this paper · explained in 2 episodes
• Regularization — 3× in this paper · explained in 3 episodes
• Adaptive Sampling — 2× in this paper · explained in 1 episode
• Ambient Dimension — 2× in this paper · explained in 1 episode
• Core — 2× in this paper · explained in 2 episodes
• DeepONet — 2× in this paper · explained in 2 episodes
• Initial Data — 2× in this paper · explained in 2 episodes
• Inverse Problem — 2× in this paper · explained in 2 episodes
• Loss Function — 2× in this paper · explained in 2 episodes
• Physics-Informed Neural Networks (PINNs) — 2× in this paper · explained in 6 episodes
• Trace — 2× in this paper · explained in 2 episodes
• VC Dimension — 2× in this paper · explained in 2 episodes
• Elias — 1× in this paper · explained in 2 episodes
• Fidelity — 1× in this paper · explained in 2 episodes
• Generalization Bounds — 1× in this paper · explained in 2 episodes
• Hilbert Space — 1× in this paper · explained in 2 episodes
• Minimax Optimal Rates — 1× in this paper · explained in 2 episodes
• PDE residual — 1× in this paper · explained in 2 episodes
• Robustness — 1× in this paper · explained in 5 episodes
• Sparsity — 1× in this paper · explained in 4 episodes
• Verification — 1× in this paper · explained in 2 episodes

Sources

• Deep ReLU network approximation of functions on a manifold

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