Data-driven approaches coupled with physical knowledge are powerful techniques to model systems. The goal of such models is to efficiently solve for the underlying field by combining measurements with known physical laws. As many systems contain unknown elements, such as missing parameters, noisy data, or incomplete physical laws, this is widely approached as an uncertainty quantification problem. The common techniques to handle all the variables typically depend on the numerical scheme used to approximate the posterior, and it is desirable to have a method which is independent of any such discretization. Information field theory (IFT) provides the tools necessary to perform statistics over fields that are not necessarily Gaussian. We extend IFT to physics-informed IFT (PIFT) by encoding the functional priors with information about the physical laws which describe the field. The posteriors derived from this PIFT remain independent of any numerical scheme and can capture multiple modes, allowing for the solution of problems which are ill-posed. We demonstrate our approach through an analytical example involving the Klein-Gordon equation. We then develop a variant of stochastic gradient Langevin dynamics to draw samples from the joint posterior over the field and model parameters. We apply our method to numerical examples with various degrees of model-form error and to inverse problems involving nonlinear differential equations. As an addendum, the method is equipped with a metric which allows the posterior to automatically quantify model-form uncertainty. Because of this, our numerical experiments show that the method remains robust to even an incorrect representation of the physics given sufficient data. We numerically demonstrate that the method correctly identifies when the physics cannot be trusted, in which case it automatically treats learning the field as a regression problem.

翻译：数据驱动方法与物理知识相结合是建模系统的强大技术。这类模型的目标是通过结合测量数据与已知物理定律，高效求解潜在场。由于许多系统包含未知要素（如缺失参数、含噪数据或不完整的物理定律），这通常被视为一个不确定性量化问题。处理所有变量的常用技术依赖于近似后验所使用的数值格式，且理想情况下应有一种独立于任何此类离散化的方法。信息场理论（IFT）提供了对未必为高斯分布的场进行统计推断的必要工具。我们通过将描述场的物理定律信息编码到函数先验中，将IFT扩展为物理信息IFT（PIFT）。由此得到的后验分布保持独立于任何数值格式，并能够捕获多模态分布，从而解决不适定问题。我们通过涉及克莱因-戈登方程的解析示例展示了该方法。随后，我们发展了一种随机梯度朗之万动力学的变体，以从场和模型参数的联合后验分布中采样。我们将该方法应用于具有不同程度模型形式误差的数值示例以及涉及非线性微分方程的逆问题。此外，该方法配备了一种度量标准，使得后验能够自动量化模型形式不确定性。因此，我们的数值实验表明，在数据充足的情况下，即使物理表示不正确，该方法仍能保持鲁棒性。我们通过数值验证表明，该方法能正确识别物理定律不可信的情况，此时它会自动将场的学习视为回归问题处理。