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。此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元素（如缺失参数、噪声数据或不完整的物理定律），这通常被视作不确定性量化问题。处理所有变量的常见技术通常依赖于用于近似后验的数值方案，而理想情况是拥有一种独立于任何此类离散化的方法。信息场理论（IFT）提供了对非高斯场进行统计所需的工具。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扩展为基于物理信息的信息场理论（PIFT）。由此得到的PIFT后验保持对数值方案的独立性，并能捕获多模态分布，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，我们开发了随机梯度朗之万动力学的一种变体，从场和模型参数的联合后验中采样。我们将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反问题。此外，该方法配备了一种度量指标，使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，即使物理表示存在错误，只要数据量充足，该方法仍保持鲁棒性。我们通过数值验证了该方法能正确识别物理不可信的情况，此时它会自动将场的求解视为回归问题。