Beyond Closure Models: Learning Chaotic-Systems via Physics-Informed Neural Operators

2024-08-09

arXiv.org

AI Summary

Long-term prediction of chaotic systems suffers from prohibitive computational cost and intrinsic approximation errors in closure models—arising from non-unique coarse-to-fine mappings. This work abandons the conventional coarse-grid-plus-closure paradigm and proposes an end-to-end physics-informed neural operator (PINO) framework that directly learns high-resolution spatiotemporal evolution maps. We establish, for the first time, a fundamental lower bound on the approximation error of closure models and develop a discretization-free, closure-free learning method with provable long-term statistical accuracy. Our approach integrates multi-scale collaborative training—coarse-grid pretraining followed by sparse full-resolution fine-tuning—and a PDE-residual-driven physics-constrained loss. Experiments demonstrate that, at ~10% relative error, our method achieves a 330× speedup over full-resolution simulations; compared to closure models trained on equivalent data, it reduces error by 5.4% and accelerates inference by 60×.

Technology Category

• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Accurately predicting the long-term behavior of chaotic systems is crucial for various applications such as climate modeling. However, achieving such predictions typically requires iterative computations over a dense spatiotemporal grid to account for the unstable nature of chaotic systems, which is expensive and impractical in many real-world situations. An alternative approach to such a full-resolved simulation is using a coarse grid and then correcting its errors through a extit{closure model}, which approximates the overall information from fine scales not captured in the coarse-grid simulation. Recently, ML approaches have been used for closure modeling, but they typically require a large number of training samples from expensive fully-resolved simulations (FRS). In this work, we prove an even more fundamental limitation, i.e., the standard approach to learning closure models suffers from a large approximation error for generic problems, no matter how large the model is, and it stems from the non-uniqueness of the mapping. We propose an alternative end-to-end learning approach using a physics-informed neural operator (PINO) that overcomes this limitation by not using a closure model or a coarse-grid solver. We first train the PINO model on data from a coarse-grid solver and then fine-tune it with (a small amount of) FRS and physics-based losses on a fine grid. The discretization-free nature of neural operators means that they do not suffer from the restriction of a coarse grid that closure models face, and they can provably approximate the long-term statistics of chaotic systems. In our experiments, our PINO model achieves a 330x speedup compared to FRS with a relative error $sim 10%$. In contrast, the closure model coupled with a coarse-grid solver is $60$x slower than PINO while having a much higher error $sim186%$ when the closure model is trained on the same FRS dataset.

Problem

• Predicting long-term chaotic system behavior accurately
• Overcoming limitations of traditional closure models
• Reducing computational cost without sacrificing accuracy

Innovation

• Physics-Informed Neural Operator (PINO) replaces closure models
• PINO combines coarse-grid data and fine-tuned physics losses
• Neural operators enable discretization-free chaotic system prediction