Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations
Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.
Code (0)
등록된 구현이 없습니다.
Tasks
Bayesian InferenceSimilar Papers 제목 키워드 기반
Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physica…
Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data
In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources from observations of system state is vit…
Operator learningUncertainty QuantificationMulti-modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimation
As data collection and simulation capabilities advance, multi-modal learning, the task of learning from multiple modalities and sources of data, is becoming an increasingly important area of research. Surrogate models th…
Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution
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) sol…
Decision MakingDeep LearningUncertainty QuantificationAutoinverse: Uncertainty Aware Inversion of Neural Networks
Neural networks are powerful surrogates for numerous forward processes. The inversion of such surrogates is extremely valuable in science and engineering. The most important property of a successful neural inverse method…