Entropy stable conservative flux form neural networks

2024-11-04

arXiv.org

AI Summary

Long-term prediction of hyperbolic conservation laws—particularly those exhibiting shocks—under noisy and sparse observations suffers from a fundamental trade-off between numerical stability and physical conservation. Method: We propose a neural network framework formulated in entropy-stable, conservative flux form. It is the first to rigorously enforce both entropy stability and exact physical conservation at the architectural level; incorporates the entropy-stable Kurganov–Tadmor numerical flux discretization; integrates a physics-constrained slope limiter as an intrinsic denoising mechanism, enabling prior-free long-term shock speed prediction; and employs unsupervised dynamical regularization. Results: The method achieves stable, long-duration simulations across diverse conservation law problems; reduces shock position and speed prediction errors by over 40% relative to baseline models; and generalizes to unseen time horizons without requiring temporal labels.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Responsible Web: Consent frameworks and practices on the web

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, {it without} oracle knowledge of later-time profiles in the training data.

Problem

• Develop entropy-stable neural network for conservation laws
• Ensure accuracy in noisy, sparse, smooth, discontinuous data
• Predict shock speeds without later-time profile knowledge

Innovation

• Entropy-stable neural network with KT scheme
• Slope limiting for denoising and accuracy
• Predicts shock speeds without later-time data