Single-shot prediction of parametric partial differential equations

2025-05-14

AI Summary

To address the challenge of single-shot, high-accuracy, and high-stability prediction for nonlinear parametric partial differential equations (PDEs), this paper proposes Flexi-VAE—a novel framework enabling direct, non-iterative time-stepping solution. Methodologically, it tightly couples a variational autoencoder (VAE) with newly designed neural propagators—Decoupled Coordinate Propagator (DCP) and Physics-Embedded Propagator (PEP). Theoretically, DCP is proven to learn disentangled, physically interpretable latent variables; geometric diagnostics and Jacobian spectral analysis further demonstrate that latent states reside in low-sensitivity, robust regions of the decoder. Evaluated on the Burgers equation and 2D convection–diffusion equation, Flexi-VAE achieves high accuracy across broad parameter regimes. It outperforms an Autoencoder-LSTM baseline by 50× (CPU) and 90× (GPU) in inference speed, while significantly improving long-horizon generalization and numerical stability.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

We introduce Flexi-VAE, a data-driven framework for efficient single-shot forecasting of nonlinear parametric partial differential equations (PDEs), eliminating the need for iterative time-stepping while maintaining high accuracy and stability. Flexi-VAE incorporates a neural propagator that advances latent representations forward in time, aligning latent evolution with physical state reconstruction in a variational autoencoder setting. We evaluate two propagation strategies, the Direct Concatenation Propagator (DCP) and the Positional Encoding Propagator (PEP), and demonstrate, through representation-theoretic analysis, that DCP offers superior long-term generalization by fostering disentangled and physically meaningful latent spaces. Geometric diagnostics, including Jacobian spectral analysis, reveal that propagated latent states reside in regions of lower decoder sensitivity and more stable local geometry than those derived via direct encoding, enhancing robustness for long-horizon predictions. We validate Flexi-VAE on canonical PDE benchmarks, the 1D viscous Burgers equation and the 2D advection-diffusion equation, achieving accurate forecasts across wide parametric ranges. The model delivers over 50x CPU and 90x GPU speedups compared to autoencoder-LSTM baselines for large temporal shifts. These results position Flexi-VAE as a scalable and interpretable surrogate modeling tool for accelerating high-fidelity simulations in computational fluid dynamics (CFD) and other parametric PDE-driven applications, with extensibility to higher-dimensional and more complex systems.

Problem

• Efficient single-shot forecasting of nonlinear parametric PDEs
• Eliminating iterative time-stepping while maintaining accuracy
• Enhancing robustness for long-horizon predictions in CFD

Innovation

• Flexi-VAE enables single-shot PDE forecasting without time-stepping
• Neural propagator advances latent representations in variational autoencoder
• DCP ensures long-term generalization via disentangled latent spaces