Solving inverse-PDE problems with physics-aware neural networks
This addresses the problem of data efficiency and accuracy in inverse-PDE tasks for computational mathematics and engineering, though it appears incremental by integrating existing techniques in a novel way.
The paper tackles inverse problems for partial differential equations (PDEs) by proposing a composite framework that blends neural networks with hard-coded PDE solver layers to discover unknown fields, demonstrating its applicability in recovering diffusion coefficients in Poisson and Burgers' equations with robustness to noise.
We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers in semantic autoencoders. Our design brings together techniques of computational mathematics, machine learning and pattern recognition under one umbrella to incorporate domain-specific knowledge and physical constraints to discover the underlying hidden fields. The network is explicitly aware of the governing physics through a hard-coded PDE solver layer in contrast to most existing methods that incorporate the governing equations in the loss function or rely on trainable convolutional layers to discover proper discretizations from data. This subsequently focuses the computational load to only the discovery of the hidden fields and therefore is more data efficient. We call this architecture Blended inverse-PDE networks (hereby dubbed BiPDE networks) and demonstrate its applicability for recovering the variable diffusion coefficient in Poisson problems in one and two spatial dimensions, as well as the diffusion coefficient in the time-dependent and nonlinear Burgers' equation in one dimension. We also show that this approach is robust to noise.