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We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces it to be a solution of the underlying PDE. We validate the model on simulated data generated by the known PDEs and added Gaussian noise, and we study our method under different levels of noise. We also compare the error of our method with a Cramer-Rao lower bound for an ordinary differential equation. Our results indicate that our method outperforms other methods in estimating PDEs, especially in the low signal-to-noise regime.
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted directly.
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already been used s
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearit
Machine learning of partial differential equations from data is a potential breakthrough to solve the lack of physical equations in complex dynamic systems, but because numerical differentiation is ill-posed to noise data, noise has become the bigges
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well a