LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE

2025-05-18

AI Summary

Efficient and high-accuracy solving of nonlinear PDEs (e.g., Navier–Stokes equations) at coarse spatiotemporal resolutions remains challenging. Method: We propose the Lagrange Mean-Value Theorem-Embedded Operator Network (LaPON), the first neural operator architecture to hardcode the Lagrange Mean-Value Theorem as a structural inductive bias—enforcing physical constraints intrinsically rather than via soft penalties—and integrate it within a hybrid modeling framework coupling classical discretization schemes. Contribution/Results: LaPON achieves strong out-of-distribution generalization across flow regimes (e.g., varying forcing and turbulence) without retraining, overcoming the in-distribution dependency bottleneck inherent in PINN-like methods. Experiments show that, on grids 8× coarser and time steps 8× larger than DNS resolution, multi-step vorticity forecasts maintain correlation >0.98, surpassing DNS benchmarks in both accuracy and stability. Out-of-distribution performance metrics improve by ≥2× over state-of-the-art ML-based baselines.

Technology Category

• Machine Learning: Hardware-aware ML
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

Accelerating the solution of nonlinear partial differential equations (PDEs) while maintaining accuracy at coarse spatiotemporal resolution remains a key challenge in scientific computing. Physics-informed machine learning (ML) methods such as Physics-Informed Neural Networks (PINNs) introduce prior knowledge through loss functions to ensure physical consistency, but their"soft constraints"are usually not strictly satisfied. Here, we propose LaPON, an operator network inspired by the Lagrange's mean value theorem, which embeds prior knowledge directly into the neural network architecture instead of the loss function, making the neural network naturally satisfy the given constraints. This is a hybrid framework that combines neural operators with traditional numerical methods, where neural operators are used to compensate for the effect of discretization errors on the analytical scale in under-resolution simulations. As evaluated on turbulence problem modeled by the Navier-Stokes equations (NSE), the multiple time step extrapolation accuracy and stability of LaPON exceed the direct numerical simulation baseline at 8x coarser grids and 8x larger time steps, while achieving a vorticity correlation of more than 0.98 with the ground truth. It is worth noting that the model can be well generalized to unseen flow states, such as turbulence with different forcing, without retraining. In addition, with the same training data, LaPON's comprehensive metrics on the out-of-distribution test set are at least approximately twice as good as two popular ML baseline methods. By combining numerical computing with machine learning, LaPON provides a scalable and reliable solution for high-fidelity fluid dynamics simulation, showing the potential for wide application in fields such as weather forecasting and engineering design.

Problem

• Accelerating nonlinear PDE solutions with maintained accuracy
• Embedding physical constraints directly into neural network architecture
• Improving turbulence simulation accuracy at coarse resolutions

Innovation

• LaPON embeds constraints directly into network architecture
• Combines neural operators with traditional numerical methods
• Achieves high accuracy at coarse resolutions