Physics-constrained convolutional neural networks for inverse problems
in spatiotemporal partial differential equations
- URL: http://arxiv.org/abs/2401.10306v2
- Date: Thu, 1 Feb 2024 15:05:43 GMT
- Title: Physics-constrained convolutional neural networks for inverse problems
in spatiotemporal partial differential equations
- Authors: Daniel Kelshaw, Luca Magri
- Abstract summary: We propose a physics-constrained convolutional neural network (PCCNN) to solve two types of inverse problems in partial differential equations (PDEs)
We analyse both linear nonlinear convection-diffusion equations and the Navier-Stokes equations, which govern the 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-temporal functions.
- Score: 5.0401589279256065
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 from the biased data the true state, which is the solution of the PDE.
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
simple 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 biased
data for the turbulent flow. We reconstruct the spatiotemporal chaotic solution
on a high-resolution grid from only 2\% 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.
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