Entropy stable conservative flux form neural networks
This work addresses the challenge of developing stable and accurate neural networks for conservation law problems in computational physics, representing an incremental improvement by combining existing numerical schemes with data-driven approaches.
The researchers tackled the problem of integrating classical numerical conservation laws into data-driven frameworks by proposing an entropy-stable conservative flux form neural network (CFN) that uses slope limiting as a denoising mechanism. The result is a model that achieves stability and conservation while maintaining accuracy over extended time domains and successfully predicting shock propagation speeds without oracle knowledge of later-time profiles in training data.
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, {\it without} oracle knowledge of later-time profiles in the training data.