Single-shot prediction of parametric partial differential equations

TL;DR

Flexi-VAE is a novel neural framework for single-shot, accurate, and stable forecasting of nonlinear parametric PDEs, significantly speeding up predictions while maintaining physical interpretability.

Contribution

The paper introduces Flexi-VAE, a new variational autoencoder-based method with neural propagators for efficient, long-term parametric PDE forecasting without iterative time-stepping.

Findings

Flexi-VAE achieves over 50x CPU and 90x GPU speedups compared to LSTM baselines.

DCP propagator provides superior long-term generalization and disentangled latent spaces.

Latent states are more stable and less sensitive to decoder variations, improving robustness.

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.