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

arXiv: https://arxiv.org/abs/2401.10306 · HTML: https://mlsanity.com/paper/2401.10306/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 40 references that are in the map.
- First posted on arXiv 18 Jan 2024.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Deep Learning for Single Image Super-Resolution: A Brief Review](https://mlsanity.com/paper/1808.03344/) (2018)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Physics-informed deep-learning applications to experimental fluid mechanics](https://mlsanity.com/paper/2203.15402/) (2022)
- [Real-time thermoacoustic data assimilation](https://mlsanity.com/paper/2106.06409/) (2021)
- [Uncovering solutions from data corrupted by systematic errors: A physics-constrained convolutional neural network approach](https://mlsanity.com/paper/2306.04600/) (2023)
- [Super-resolving sparse observations in partial differential equations: A physics-constrained convolutional neural network approach](https://mlsanity.com/paper/2306.10990/) (2023)

## Built on by (most cited first, 2 of 2)

- [Reconstructing unsteady flows from sparse, noisy measurements with a physics-constrained convolutional neural network](https://mlsanity.com/paper/2409.00260/) (2024)
- [Efficient Adaptation of ROMs for Unsteady Flows Using Data Assimilation](https://mlsanity.com/paper/2602.23188/) (2026)

## Abstract (arXiv)

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.