Solving partial differential equations (PDEs) and their inverse problems
using Physics-informed neural networks (PINNs) is a rapidly growing approach in
the physics and machine learning community. Although several architectures
exist for PINNs that work remarkably in practice, our theoretical understanding
of their performances is somewhat limited. In this work, we study the behavior
of a Bayesian PINN estimator of the solution of a PDE from $n$ independent
noisy measurement of the solution. We focus on a class of equations that are
linear in their parameters (with unknown coefficients $\theta_\star$). We show
that when the partial differential equation admits a classical solution (say
$u_\star$), differentiable to order $\beta$, the mean square error of the
Bayesian posterior mean is at least of order $n^{-2\beta/(2\beta + d)}$.
Furthermore, we establish a convergence rate of the linear coefficients of
$\theta_\star$ depending on the order of the underlying differential operator.
Last but not least, our theoretical results are validated through extensive
simulations.

使用物理约束的神经网络和贝叶斯后验均值估计的方法在物理和机器学习领域得到了广泛的应用，本研究探讨了这些方法在解决偏微分方程及其反问题时的性能和收敛性。