Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs

Abstract

Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural networks provide fast inference and generalize but either require large amounts of training data or a physics-constrained loss based on finite differences that can lead to inaccuracies and discretization artifacts. We leverage the advantages of both of these approaches by using Hermite spline kernels in order to continuously interpolate a grid-based state representation that can be handled by a CNN. This allows for training without any precomputed training data using a physics-informed loss function only and provides fast, continuous solutions that generalize to unseen domains. We demonstrate the potential of our method at the examples of the incompressible Navier-Stokes equation and the damped wave equation. Our models are able to learn several intriguing phenomena such as Karman vortex streets, the Magnus effect, Doppler effect, interference patterns and wave reflections. Our quantitative assessment and an interactive real-time demo show that we are narrowing the gap in accuracy of unsupervised ML based methods to industrial solvers for computational fluid dynamics (CFD) while being orders of magnitude faster.

Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN

“…This approach facilitates efficient training of implicit neural networks, producing continuous solutions that can be evaluated at any spatiotemporal point. However, implicit neural networks tend to lack generalization capabilities in novel domains and often necessitate network retraining [11].…”

“…Researchers have been paying more attention to PINNs lately, and they are gradually applying them to a wider range of research fields [10]. Although PINNs can be trained with little to no ground truth data, they often fail to effectively generalize to domains that were not encountered during training [11]. However, CNN can learn the inherent laws of the data and thus have better generalization ability for new data.…”

LordNet: Learning to Solve Parametric Partial Differential Equations without Simulated Data

“…Geneva and Zabaras [2020] introduced the use of physics-constrained framework to achieve the data-free training in the case of Burgers's equations. Wandel et al [2020Wandel et al [ , 2021 proposed the physics-constrained loss based on the discretization of the PDE but they are specific to certain PDEs, like Navier-Stokes equation. These physics-constrained losses can be seen as special cases of the MSR loss.…”

Accelerating Atmospheric Gravity Wave Simulations using Machine Learning

“…The most explored of these can be divided into two categories: physics-informed neural networks (PINN, e.g., Maziar et al, 2019;Wandel et al, 2022) and neural operators (NOs, e.g., Lu et al, 2019;Li et al, 2020;Xiong et al, 2023). PINN uses a neural network as the solution function and optimizes a loss function to minimize violation of the given equation.…”