用低秩微调加速贝叶斯偏微分方程反问题的不确定性量化。
Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptation
- 上下层协同：上层用哈密顿蒙特卡洛采样，下层用低秩适配微调神经网络。
- 无需高维权重采样或伴随方程，计算效率显著提升。
- 适用于肿瘤生长等复杂模型，兼具精度与高效性，适合科学计算场景。
偏微分方程反问题中的不确定性量化在众多应用中至关重要。科学机器学习与人工智能使模型组件能够数据驱动地学习，同时保持物理结构，并具备应对新兴成像技术与临床洞察所需的可扩展性与适应性。我们提出一种用于偏微分方程贝叶斯推断的双层局部算子学习框架（B-BiLO）。上层通过哈密顿蒙特卡洛从后验分布中采样参数，下层则利用低秩适配（LoRA）微调神经网络，以局部逼近解算子。B-BiLO 实现了无需合成数据或伴随方程的基于梯度的高效采样，避免了贝叶斯神经网络中高维权重空间的采样问题，通过确定性优化权重实现高效推断。我们分析了下层近似优化带来的误差及其对后验精度的影响。数值实验涵盖多种偏微分方程模型，包括肿瘤生长模型，结果表明 B-BiLO 能实现准确且高效的不确定性量化。
原文摘要 · Abstract (English)
Uncertainty quantification in PDE inverse problems is essential in many applications. Scientific machine learning and AI enable data-driven learning of model components while preserving physical structure, and provide the scalability and adaptability needed for emerging imaging technologies and clinical insights. We develop a Bilevel Local Operator Learning framework for Bayesian inference in PDEs (B-BiLO). At the upper level, we sample parameters from the posterior via Hamiltonian Monte Carlo, while at the lower level we fine-tune a neural network via low-rank adaptation (LoRA) to approximate the solution operator locally. B-BiLO enables efficient gradient-based sampling without synthetic data or adjoint equations and avoids sampling in high-dimensional weight space, as in Bayesian neural networks, by optimizing weights deterministically. We analyze errors from approximate lower-level optimization and establish their impact on posterior accuracy. Numerical experiments across PDE models, including tumor growth, demonstrate that B-BiLO achieves accurate and efficient uncertainty quantification.
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