We present the application of the physics-informed neural network (PINN) approach in Bayesian formulation. We have adopted the Bayesian neural network framework to obtain posterior densities from Laplace approximation. For each model or fit, the evidence is computed, which is a measure that classifies the hypothesis. The optimal solution is the one with the highest value of evidence. We have proposed a modification of the Bayesian algorithm to obtain hyperparameters of the model. We have shown that within the Bayesian framework, one can obtain the relative weights between the boundary and equation contributions to the total loss. Presented method leads to predictions comparable to those obtained by sampling from the posterior distribution within the Hybrid Monte Carlo algorithm (HMC). We have solved heat, wave, and Burger's equations, and the results obtained are in agreement with the exact solutions, demonstrating the effectiveness of our approach. In Burger's equation problem, we have demonstrated that the framework can combine information from differential equations and potential measurements. All solutions are provided with uncertainties (induced by the model's parameter dependence) computed within the Bayesian framework.

翻译：本文介绍了物理信息神经网络（PINN）在贝叶斯框架下的应用。我们采用贝叶斯神经网络框架通过拉普拉斯近似获取后验密度。针对每个模型或拟合，计算证据值——这一度量可对假设进行分类。最优解对应证据值最大的结果。我们提出了贝叶斯算法的改进方案以获取模型超参数，并证明在贝叶斯框架下可确定边界与方程贡献项在总损失中的相对权重。所提方法的预测能力与混合蒙特卡洛算法（HMC）从后验分布采样的结果相当。我们求解了热传导方程、波动方程和伯格斯方程，所得结果与精确解一致，验证了方法的有效性。在伯格斯方程问题中，我们证明了该框架可融合微分方程与潜在测量值的信息。所有解均包含贝叶斯框架下计算的不确定性（由模型参数依赖关系引发）。