Predicting Change, Not States: An Alternate Framework for Neural PDE Surrogates

2024-12-17

Computer Methods in Applied Mechanics and Engineering

AI Summary

This work addresses the poor long-term stability and generalization of neural PDE surrogate models that directly predict system states. We propose a novel paradigm: predicting time derivatives followed by ODE integration. Methodologically, we build upon the neural ODE framework, jointly optimizing a physics-informed loss and a self-supervised temporal differencing regularization, enabling implicit differentiation and adaptive-step integration. Our key contribution lies in decoupling physical constraints from data fitting—shifting the learning objective from “predicting states” to “predicting dynamics”—thereby substantially mitigating error accumulation. Evaluated on canonical PDE benchmarks—including Burgers’ and Navier–Stokes equations—our approach achieves a 38% reduction in average relative error, improves extrapolation capability by 2.1×, and accelerates training convergence by 1.6×, while maintaining high accuracy, strong numerical stability, and flexible inference.

Technology Category

• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP

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
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Problem

• Predict temporal derivatives instead of next state for PDEs
• Improve neural surrogate accuracy and stability
• Enable flexible time-stepping during inference

Innovation

• Predict temporal derivatives instead of next states
• Use ODE integrator for flexible time-stepping
• Improve accuracy and stability in fine discretization