We propose a randomized physics-informed neural network (PINN) or rPINN method for uncertainty quantification in inverse partial differential equation (PDE) problems with noisy data. This method is used to quantify uncertainty in the inverse PDE PINN solutions. Recently, the Bayesian PINN (BPINN) method was proposed, where the posterior distribution of the PINN parameters was formulated using the Bayes' theorem and sampled using approximate inference methods such as the Hamiltonian Monte Carlo (HMC) and variational inference (VI) methods. In this work, we demonstrate that HMC fails to converge for non-linear inverse PDE problems. As an alternative to HMC, we sample the distribution by solving the stochastic optimization problem obtained by randomizing the PINN loss function. The effectiveness of the rPINN method is tested for linear and non-linear Poisson equations, and the diffusion equation with a high-dimensional space-dependent diffusion coefficient. The rPINN method provides informative distributions for all considered problems. For the linear Poisson equation, HMC and rPINN produce similar distributions, but rPINN is on average 27 times faster than HMC. For the non-linear Poison and diffusion equations, the HMC method fails to converge because a single HMC chain cannot sample multiple modes of the posterior distribution of the PINN parameters in a reasonable amount of time.

翻译：我们提出了一种随机物理信息神经网络（PINN）或 rPINN 方法，用于含噪声数据的反演偏微分方程（PDE）问题中的不确定性量化。该方法用于量化反演 PDE PINN 解中的不确定性。最近，贝叶斯 PINN（BPINN）方法被提出，其中使用贝叶斯定理构建了 PINN 参数的后验分布，并使用哈密顿蒙特卡洛（HMC）和变分推断（VI）等近似推断方法进行采样。在这项工作中，我们证明了 HMC 对于非线性反演 PDE 问题无法收敛。作为 HMC 的替代方案，我们通过求解随机化 PINN 损失函数得到的随机优化问题来采样该分布。rPINN 方法的有效性在线性和非线性泊松方程以及具有高维空间相关扩散系数的扩散方程上进行了测试。rPINN 方法为所有考虑的问题提供了信息丰富的分布。对于线性泊松方程，HMC 和 rPINN 产生相似的分布，但 rPINN 平均比 HMC 快 27 倍。对于非线性泊松方程和扩散方程，HMC 方法无法收敛，因为单个 HMC 链无法在合理的时间内采样 PINN 参数后验分布的多个模态。