Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution

TL;DR

This work introduces LE-PDE-UQ, a latent-space framework that quantifies predictive uncertainty for both forward evolution and inverse optimization of time-dependent PDEs. By encoding the system state and its uncertainty into latent vectors and evolving them with a dedicated latent dynamics model, the approach enables accurate long-horizon uncertainty propagation without repeated sampling. Through a 2D Navier–Stokes turbulence dataset, LE-PDE-UQ outperforms Bayes layers, dropout, and deep ensembles in both uncertainty calibration and predictive accuracy, with ablations highlighting the critical roles of latent evolution and uncertainty propagation. The method also facilitates efficient inverse optimization by operating in a low-dimensional latent space and using deep ensembles for uncertainty, achieving physically plausible solutions with robust uncertainty estimates. Overall, LE-PDE-UQ advances trustworthy, scalable UQ for neural PDE surrogates, offering practical impact for scientific computing and engineering design.

Abstract

Deep learning-based surrogate models have demonstrated remarkable advantages over classical solvers in terms of speed, often achieving speedups of 10 to 1000 times over traditional partial differential equation (PDE) solvers. However, a significant challenge hindering their widespread adoption in both scientific and industrial domains is the lack of understanding about their prediction uncertainties, particularly in scenarios that involve critical decision making. To address this limitation, we propose a method that integrates efficient and precise uncertainty quantification into a deep learning-based surrogate model. Our method, termed Latent Evolution of PDEs with Uncertainty Quantification (LE-PDE-UQ), endows deep learning-based surrogate models with robust and efficient uncertainty quantification capabilities for both forward and inverse problems. LE-PDE-UQ leverages latent vectors within a latent space to evolve both the system's state and its corresponding uncertainty estimation. The latent vectors are decoded to provide predictions for the system's state as well as estimates of its uncertainty. In extensive experiments, we demonstrate the accurate uncertainty quantification performance of our approach, surpassing that of strong baselines including deep ensembles, Bayesian neural network layers, and dropout. Our method excels at propagating uncertainty over extended auto-regressive rollouts, making it suitable for scenarios involving long-term predictions. Our code is available at: https://github.com/AI4Science-WestlakeU/le-pde-uq.

Figures (15)

• Figure 1: Schematic representation of the LE-PDE-UQ framework. In the forward mode(green), LE-PDE-UQ evolves the dynamics in the global latent space. In the inverse optimization mode (red), it optimizes the parameter p(e.g., the boundary) by unrolling the latent vectors. The compressed latent vectors and dynamics can significantly speed up both modes. The latent evolution model g deterministically predicts a global latent vector z encoding the state and a global latent vector $Z_{\sigma}$ encoding the uncertainty. On demand, they are decoded into the predicted state and the predicted uncertainty, respectively.
• Figure 2: Visualization of forward prediction results of LE-PDE-UQ on 2D Navier-Stokes turbulent flow dataset. The figure predicts the fluid state from 11-20 steps using the actual fluid state data from 1-10 steps. In this context, (a) represents the actual fluid state from 11-20 steps, while (b) indicates the fluid state predicted by LE-PDE-UQ. (c) represents the absolute error between (a) and (b), and (d) represents the uncertainty quantification results obtained by LE-PDE-UQ.
• Figure 3: The ordered prediction intervals and average calibration of LE-PDE-UQ.
• Figure 4: The ordered prediction intervals and average calibration of NoLatent (ensemble, with $\sigma$).
• Figure 5: The ordered Prediction Intervals and Average Calibration Plot Analysis for Latent full + Deterministic case.