Beyond Closure Models: Learning Chaotic-Systems via Physics-Informed Neural Operators (2408.05177v4)

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 \textit{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.

• The paper demonstrates that Physics-Informed Neural Operators overcome traditional closure models’ limitations in capturing chaotic system dynamics.
• It employs a training strategy using coarse-grid data followed by fine-tuning with high-resolution simulations to achieve 120x speedup and about 5% error.
• The study reveals practical and theoretical advances, integrating functional Liouville flow principles to provide robust, efficient chaotic system modeling.

Summary of the Paper: Beyond Closure Models: Learning Chaotic-Systems via Physics-Informed Neural Operators

The paper "Beyond Closure Models: Learning Chaotic-Systems via Physics-Informed Neural Operators" addresses the significant challenge of accurately predicting long-term behavior in chaotic systems, like climate dynamics, using ML techniques. Traditional methods often rely on fine-grid numerical simulations to capture intricate dynamics, but these approaches are computationally prohibitive. Coarse-grid simulations married with closure models offer a practical alternative, albeit with inherent limitations. This paper critiques standard closure models and proposes an innovative method leveraging Physics-Informed Neural Operators (PINO) for more robust approximations.

Critique of Current Closure Models

Chaotic systems, with their sensitivity to initial conditions and perturbations, require accurate modeling to predict long-term statistics. Traditionally, closure models have been deployed to approximate the influences of unresolved scales not captured by coarse-grain simulations. However, the paper highlights critical shortcomings of these models, including significant approximation errors due to the non-uniqueness of mapping from coarse-grid data to fine-grid predictions. Specifically, the mapping of closure models is not well-defined; hence, the model tends to average multiple plausible outputs into potentially inaccurate predictions.

Introduction of Physics-Informed Neural Operators

Departing from closure models, the authors propose a method using PINO, which circumvents the non-uniqueness problem. PINO is demonstrated as a superior approach due to its discrete-agnostic nature, allowing it to learn functions across various resolutions without the constraints of traditional numerical grids.

Methodology:

1. Training on Coarse Data: Initially, the PINO model is trained using data from a coarse-grid simulator.
2. Fine-Tuning with Limited Data: The model is fine-tuned with a relatively small amount of data from fully-resolved simulations (FRS) and additional physics-consistent losses to model fine-grid dynamics accurately.
3. End-to-End Function Learning: By adopting a neural operator strategy, the PINO framework learns to map entire functions rather than discrete grid points, reducing dimensionality-induced errors.

Theoretical and Practical Implications

Theoretically, the paper offers a new perspective by employing the framework of functional Liouville flow in the paper of chaotic systems, providing analytical insight into the deficiencies of closure models. The neural operator model promises a more universally applicable approximation method. The method benefits from a solid theoretical foundation affirming that PINO can estimate long-term statistics with bounded errors, even when minor deviations from perfect predictions exist.

Practically, the PINO model shows a massive computational speedup—120x faster than traditional fine-resolved simulations while maintaining accuracy within a relative error of about 5%. In turbulence and fluid dynamics problems, the approach offers vast improvements over both traditional closure models (which prove much slower and less accurate) and ML-based closure approaches when severely data-constrained.

Future Directions

The work opens several avenues for future exploration in artificial intelligence and chaotic system modeling:

• Multifidelity Data Learning: Further integrating PINO into broader contexts for learning from multiple fidelity data sets, potentially reducing the reliance on high-fidelity simulations even further.
• Generalization Studies: Extending the application of PINO to other classes of PDEs and dynamical systems, which might exhibit different chaotic properties.
• Robustness & Scalability: Investigating the robustness of PINO under various physical perturbations and scaling it to more extensive, industry-relevant problems in fluid dynamics and beyond.

In sum, the paper presents a meaningful advance in leveraging neural operators for chaotic systems, moving beyond the limitations of standard closure models, and offers compelling evidence of the efficacy and efficiency of PINO in practical applications.