Learning Generalizable Neural Operators for Inverse Problems

TL;DR

The paper tackles ill-posed inverse problems by decoupling function representation from inversion through a basis-to-basis framework. It introduces B2B^{-1}, which learns neural bases for input and output spaces and performs inversion in coefficient space, enabling deterministic, invertible, and probabilistic inverse models that can reflect uncertainty and non-uniqueness. Evaluations across six PDE benchmarks—including two novel inverse-operator datasets (Chladni plate resonance and wave scattering) and a seismic full waveform inversion task—show that probabilistic inverses are essential for strongly ill-posed tasks, while deterministic and invertible models excel on well-posed cases. The results, interpreted via forward-consistency re-simulation, demonstrate robust generalization and highlight tradeoffs between accuracy and robustness to measurement noise, guiding model selection based on problem structure and data quality. Overall, the B2B^{-1} framework offers a scalable, modular approach to principled inverse surrogate modeling with potential across engineering and geophysics domains.

Abstract

Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness.

Figures (5)

• Figure 1: B2B${}^{-1}$ inverse operator framework. Sensor measurements (1) are projected onto learned neural basis functions to produce a low-dimensional coefficient vector $\beta$ (2). The inverse neural operator maps $\beta$ to input coefficients $\alpha$ (2-4) of a second learned neural basis of the input space. This operator can be deterministic, invertible, or probabilistic; in the probabilistic case, it defines a distribution over plausible $\alpha$. We pass sampled inputs through a separate forward neural operator $T$ for re-simulation (5–6) and validation.
• Figure 2: Qualitative comparison of inverse model performance on classical PDE benchmarks. We visualize the best average representative sample across test data for each dataset (Darcy, Burgers, Elastic Plate). For probabilistic models, input reconstructions show random samples drawn from the predicted distribution. The corresponding output re-simulations, generated by passing these samples through the forward model, are shown on the right. Effective models recover plausible input functions and produce accurate output reconstructions when evaluated under the forward operator.
• Figure 3: Qualitative comparison of inverse model performance on inverse operator benchmarks. We display representative test samples for the wave scattering and Chladni datasets, highlighting the top-performing models. For probabilistic models, input reconstructions are random samples from the predicted distribution, with corresponding output re-simulations shown on the right.
• Figure 4: Qualitative comparison of inverse models on the FWI benchmark. We show representative reconstructions of subsurface velocity models (left) and corresponding seismic re-simulations (right) for the top-performing inverse models. While models generally recover large-scale structural features of the velocity field, they often fail to capture high-frequency components, resulting in smoothed re-simulated wavefields.
• Figure 5: Sensitivity of inverse models to observation noise on the elastic plate benchmark.