Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations

TL;DR

The paper addresses inverse problems for spatiotemporal PDEs by introducing a physics-constrained CNN (PC-CNN) with a time-windowing scheme to enforce governing equations without relying on automatic differentiation. It tackles two tasks: removing spatially varying bias from biased observations to recover the true PDE solution, and reconstructing high-resolution fields from sparse information, even in chaotic turbulent regimes. The approach uses a composite loss with data, physics, and constraint terms and leverages a pseudospectral discretisation with Fourier-domain physics losses, achieving accurate reconstructions and physically consistent results (e.g., correct energy spectra) where standard interpolation fails. The findings demonstrate robust performance across linear, nonlinear, and Navier–Stokes dynamics, suggesting broad applicability for solving inverse PDE problems in complex spatiotemporal systems.

Abstract

We propose a physics-constrained convolutional neural network (PC-CNN) to solve two types of inverse problems in partial differential equations (PDEs), which are nonlinear and vary both in space and time. In the first inverse problem, we are given data that is offset by spatially varying systematic error (i.e., the bias, also known as the epistemic uncertainty). The task is to uncover the true state, which is the solution of the PDE, from the biased data. In the second inverse problem, we are given sparse information on the solution of a PDE. The task is to reconstruct the solution in space with high-resolution. First, we present the PC-CNN, which constrains the PDE with a time-windowing scheme to handle sequential data. Second, we analyse the performance of the PC-CNN for uncovering solutions from biased data. We analyse both linear and nonlinear convection-diffusion equations, and the Navier-Stokes equations, which govern the spatiotemporally chaotic dynamics of turbulent flows. We find that the PC-CNN correctly recovers the true solution for a variety of biases, which are parameterised as non-convex functions. Third, we analyse the performance of the PC-CNN for reconstructing solutions from sparse information for the turbulent flow. We reconstruct the spatiotemporal chaotic solution on a high-resolution grid from only 1% of the information contained in it. For both tasks, we further analyse the Navier-Stokes solutions. We find that the inferred solutions have a physical spectral energy content, whereas traditional methods, such as interpolation, do not. This work opens opportunities for solving inverse problems with partial differential equations.

Figures (11)

• Figure 1: Inverse problems investigated in this paper. (a) Uncovering solutions from biased data. The model ${\bm {\eta_\theta}}$ is responsible for recovering the solution (true state), ${\bm u}({\bm \Omega}, t)$, from the biased data, ${\bm \zeta}(\bm{\Omega}, t)$. The bias (systematic error), ${\bm \phi}({\bm x})$, is the difference between the biased data and the solution. (b) Reconstructing a solution from sparse information. The model ${\bm {f_\theta}}$ is responsible for mapping the sparse field ${\bm u}({\bm \Omega_L}, t)$ to the high-resolution field ${\bm u}({\bm \Omega_H}, t)$. The term $\tau$ in both cases denotes the number of contiguous time-steps passed to the network, required for computing temporal derivatives. An explanation of the proposed physics-constrained convolutional neural network (PC-CNN), which is the ansatz for both mappings ${\bm \eta_{\theta}}$ and ${\bm f_{\theta}}$, is provided in Sec. \ref{['sec:pc_cnn_overview']}.
• Figure 2: Time-windowing scheme. Time-steps are first grouped into non-overlapping subsets of successive elements of length $\tau$. Each of these subsets can be taken for either training, or validation. Subsets are treated as minibatches and passed through the network to evaluate their output. The temporal derivative is then approximated using a forward-Euler approximation across adjacent time-steps.
• Figure 3: Uncovering solutions from biased data. (a) Linear convection-diffusion case $k_\phi = 3$ and $\mathcal{M}=0.5$. (b) Nonlinear convection-diffusion case with $k_\phi = 5$ and $\mathcal{M}=0.5$. (c) Two-dimensional turbulent flow case with $[k_\phi = 7$ and $\mathcal{M}=0.5$. Panel (i) shows the biased data, ${\bm \zeta}$; (ii) shows the true state, ${\bm u}$, which we wish to uncover from the biased data; (iii) shows the bias, which represents the bias (i.e., systematic error), ${\bm \phi}$; (iv) shows the network predictions, ${\bm \eta_\theta}$; and (v) shows the predicted bias, ${\bm \zeta}$ - ${\bm \eta_\theta}$.
• Figure 4: Unconvering solutions from biased data: robustness analysis through relative error, $e$. (a) Linear convection-diffusion case. (b) Nonlinear convection-diffusion case. (c) Two-dimensional turbulent flow case. Orange-bars denote results for case $\mathit{(i)}$: fixing the magnitude and varying the Rastrigin wavenumber. Blue-bars denote results for case $\mathit{(ii)}$: fixing the Rastrigin wavenumber and varying the magnitude.
• Figure 5: Uncovering solution from biased data. Temporal evolution of the two-dimensional turbulent flow with $[k_\phi = 7, \mathcal{M} = 0.5]$. $T_t$ denotes the length of the transient. (i) Biased data, ${\bm \zeta}$; (ii) true state, ${\bm u}$; (iii) predicted solution, ${\bm \eta_\theta}$; and (iv) squared error, $||{\bm \eta_\theta} - {\bm u}||^2$.