用神经网络代理模型实现带不确定性的微分方程反问题求解
Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations
- 构建基于神经网络的微分方程求解代理模型，支持不确定性量化
- 在高维参数空间下仍保持计算效率，精度媲美经典高斯过程代理
- 适用于非线性情形，适合需可靠贝叶斯推断的复杂系统研究
微分方程的反问题广泛存在于科学与工程中，目标是从噪声或不完整观测中推断未知模型参数。传统数值方法在贝叶斯框架下常因前向模型复杂且参数维度高而计算成本高昂。为此，我们提出DeepGaLA，一种用于微分方程求解器的神经网络代理模型，可提供带有不确定性的预测，缓解训练数据有限时的过度自信问题。为评估代理模型诱导后验近似的保真度，我们证明短时延迟接受马尔可夫链蒙特卡洛可作为有效诊断工具。在多种数值实验中，DeepGaLA在前向模型近似上达到与现有高斯过程代理相当的精度，且随着参数维度增加仍保持更高效率。此外，该模型能融入微分方程约束，包括非线性情形。总体表明，带有不确定性量化的神经代理模型可实现复杂系统反问题中可扩展且可靠的贝叶斯推断。
原文摘要 · Abstract (English)
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.
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