The joint work with L. Liu, T. Li, and A. Gelb Entropy stable conservative flux form neural networks , has been submitted for publication in SIAM Journal of Scientific Computing. This work uses entropy-stable scheme in learning conservation laws, extending my previous work with Z. Chen and A. Gelb Learning the dynamics for unknown hyperbolic conservation laws using deep neural networks.
Abstract: We propose an entropy-stable conservative flux form neural network (CFN) that integrates classical numerical conservation laws into a data-driven framework using the entropy-stable, second-order, and non-oscillatory Kurganov-Tadmor (KT) scheme. The proposed entropy-stable CFN uses slope limiting as a denoising mechanism, ensuring accurate predictions in both noisy and sparse observation environments, as well as in both smooth and discontinuous regions. Numerical experiments demonstrate that the entropy-stable CFN achieves both stability and conservation while maintaining accuracy over extended time domains. Furthermore, it successfully predicts shock propagation speeds in long-term simulations, without oracle knowledge of later-time profiles in the training data.