Physics informed neural network (PINN) approach in Bayesian formulation is presented. We adopt the Bayesian neural network framework formulated by MacKay (Neural Computation 4 (3) (1992) 448). The posterior densities are obtained from Laplace approximation. For each model (fit), the so-called evidence is computed. It is a measure that classifies the hypothesis. The most optimal solution has the maximal value of the evidence. The Bayesian framework allows us to control the impact of the boundary contribution to the total loss. Indeed, the relative weights of loss components are fine-tuned by the Bayesian algorithm. We solve heat, wave, and Burger's equations. The obtained results are in good agreement with the exact solutions. All solutions are provided with the uncertainties computed within the Bayesian framework.

翻译：本文提出了贝叶斯框架下的物理信息神经网络（PINN）方法。我们采用了MacKay（Neural Computation 4 (3) (1992) 448）提出的贝叶斯神经网络框架，通过拉普拉斯逼近获得后验密度。针对每个模型（拟合），计算所谓的“证据”，该度量用于对假设进行分类，最优解对应证据的最大值。贝叶斯框架使我们能够控制边界贡献对总损失的影响，即损失组件的相对权重由贝叶斯算法自动微调。我们求解了热传导方程、波动方程和伯格斯方程，所得结果与精确解高度吻合。所有解均附有在贝叶斯框架内计算的不确定性。