面向多任务与多尺度正反问题的贝叶斯物理信息神经网络自适应加权
Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems
AI总结：
本文提出一种贝叶斯物理信息神经网络的自适应加权方法，通过多任务后验分布自动调节权重，以改善多尺度正反问题的收敛性、稳定性并减少采样偏差。
摘要：
本文提出了一种对贝叶斯物理信息神经网络（BPINNs）进行自动自适应加权的新方法，并证明该方法能够稳健地处理多目标和多尺度问题。BPINNs 是一种流行的数据同化框架，结合了不确定性量化（UQ）和偏微分方程（PDE）的约束。BPINN 目标分布各项的相对权重与相应学习任务中的固有不确定性直接相关。然而，这些权重通常由人工先验设定，可能导致病态行为、稳定性问题以及任务之间的冲突，这些障碍阻碍了 BPINNs 在具有多尺度动力学的反问题中的应用。本文提出的加权策略通过考虑目标后验分布的多任务性质来自动调整权重。我们表明，该方法能够修复 BPINNs 的失效模式，并提供对最优 Pareto 前沿的高效探索。这带来了 BPINN 训练更好的收敛性和稳定性，同时减少了采样偏差。所确定的权重还携带关于任务不确定性的信息，反映数据中的噪声水平以及 PDE 模型的充分性。我们在 Sobolev 训练的数值实验中对此进行了验证，并将其与解析的 ε-最优基线以及一个多尺度 Lotka-Volterra 反问题进行了比较。最后，我们将该框架应用于一个图像修复任务和一个反问题，其中涉及复杂几何形状中不可压缩流动的潜在场恢复。
英文摘要：
In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multi-scale problems. BPINNs are a popular framework for data assimilation, combining the constraints of Uncertainty Quantification (UQ) and Partial Differential Equation (PDE). The relative weights of the BPINN target distribution terms are directly related to the inherent uncertainty in the respective learning tasks. Yet, they are usually manually set a-priori, that can lead to pathological behavior, stability concerns, and to conflicts between tasks which are obstacles that have deterred the use of BPINNs for inverse problems with multi-scale dynamics. The present weighting strategy automatically tunes the weights by considering the multi-task nature of target posterior distribution. We show that this remedies the failure modes of BPINNs and provides efficient exploration of the optimal Pareto front. This leads to better convergence and stability of BPINN training while reducing sampling bias. The determined weights moreover carry information about task uncertainties, reflecting noise levels in the data and adequacy of the PDE model. We demonstrate this in numerical experiments in Sobolev training, and compare them to analytically $ε$-optimal baseline, and in a multi-scale Lokta-Volterra inverse problem. We eventually apply this framework to an inpainting task and an inverse problem, involving latent field recovery for incompressible flow in complex geometries.
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