Adaptive operator learning for infinite-dimensional Bayesian inverse problems

TL;DR

This work tackles infinite-dimensional Bayesian inverse problems governed by PDEs by marrying an online adaptive neural-operator surrogate (DeepOnet) with Unscented Kalman Inversion (UKI). A greedy online refinement strategy selects local training points from the evolving posterior to minimize surrogate modeling error where the posterior mass is concentrated, ensuring accurate inversion with substantially reduced forward-model evaluations. The authors provide a convergence guarantee in the linear setting and demonstrate the method on Darcy flow, heat-source inversion, and a reaction-diffusion problem, achieving comparable accuracy to full-order solvers at a fraction of the cost. The approach offers a practical path toward real-time or many-query Bayesian inference in complex PDE settings, while acknowledging possible limitations for non-Gaussian posteriors or low-dimensional regimes where alternative calibration strategies may be preferable.

Abstract

The fundamental computational issues in Bayesian inverse problems (BIP) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such costs is to replace expensive model simulations with computationally efficient approximations using operator learning, motivated by recent progress in deep learning. However, using the approximated model directly may introduce a modeling error, exacerbating the already ill-posedness of inverse problems. Thus, balancing between accuracy and efficiency is essential for the effective implementation of such approaches. To this end, we develop an adaptive operator learning framework that can reduce modeling error gradually by forcing the surrogate to be accurate in local areas. This is accomplished by adaptively fine-tuning the pre-trained approximate model with training points chosen by a greedy algorithm during the posterior evaluation process. To validate our approach, we use DeepOnet to construct the surrogate and unscented Kalman inversion (UKI) to approximate the BIP solution, respectively. Furthermore, we present a rigorous convergence guarantee in the linear case using the UKI framework. The approach is tested on a number of benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction-diffusion problem. The numerical results show that our method can significantly reduce computational costs while maintaining inversion accuracy.

Figures (22)

• Figure 2.1: Diagram of the three components of DeepOnet.
• Figure 2.2: The framework of DeepOnet.
• Figure 3.1: The flowchart for our mutual learning framework. The framework begins with an offline, pre-trained operator network. The operator network combines with observations to efficiently return likelihood information for samplers to explore the parameter space, while the samplers provide useful candidate points for surrogate refinement. They can learn from and improve each other over time.
• Figure 4.1: The ground truth for in-distribution data(IDD) and out-of-distribution data (OOD) from above to below. Left: the true permeability fields $\mathbf{m}(\mathbf{x})$. Right: the pressure fields $u(\mathbf{x})$ and the corresponding $36$ equidistant observations with noise level 0.01.
• Figure 4.2: The data fitting error (left), model error (middle) and relative inversion error (right) for Example 1. Above: IDD case. Below: OOD case.

Theorems & Definitions (7)

• Theorem 2.1: yan2019adaptive1
• Theorem 3.1: Modified Unscented Transform wan2000unscented
• Lemma 3.5
• proof
• Theorem 3.6
• proof
• Remark 1