Artificial neural networks for solving ordinary and partial differential equations

By Isaac E. Lagaris, Aristidis C. Likas, Dimitrios I. Fotiadis. 1997

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

Subfields: Numerical Methods & Scientific Computing, AI for Physical Sciences

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, background): “[168, 169] explored the idea of using neural networks to solve initial and boundary value problems.”
Physics-Informed Neural Networks for Cardiac Activation Mapping (2020, methodology): “In this section we introduce a physics-informed neural network [9, 18] to interpolate activation times for cardiac electrophysiology.”
Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, methodology): “This framework has enabled the solution of forward and inverse problem in computational physics by reviving the original ideas in [16, 17] using modern software developments in reverse-mode differentiation [18] in order to automatically compute any derivatives present in the PDE operator.”
Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data (2019, methodology): “In [32], a one-hidden-layer fully-connected neural network (FC-NN) with spatial coordinates as input is trained to minimize the residual norm evaluated on a fixed grid.”
fPINNs: Fractional Physics-Informed Neural Networks (2018, background): “In particular, we concentrate on physics-informed neural networks (PINNs) [7, 46, 19, 35], which belong to the second aforementioned category.”
Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (2018, background): “This new network has the same parameters as the network representing u(x, t), albeit different activation functions due to the action of the differential operator [6, 11, 12].”
DGM: A deep learning algorithm for solving partial differential equations (2017, positive): “[28], [29], [45], [30], and [34] propose to use neural networks to solve PDEs and ODEs.”
Physics-informed deep learning for incompressible laminar flows (2020, background): “To address this fundamental challenge, physics-informed DL explicitly embed the physical laws (e.g., the governing partial differential equations (PDEs), initial/boundary conditions, etc.) into the DNN which provide rigorous constraint to the network trainable parameters within a feasible solution s”
Machine Learning for Fluid Mechanics (2019, background): “Early examples include the use of NNs to learn the solutions of ordinary and partial diﬀerential equations (Dis-sanayake & Phan-Thien 1994; Gonzalez-Garcia et al. 1998; Lagaris et al. 1998).”
Informed Machine Learning – A Taxonomy and Survey of Integrating Prior Knowledge into Learning Systems (2019, methodology): “this end, the governing differential equation is integrated into the loss function similar to Equation (1) [45].”
CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method (2021, background): “The concept of PINNs can be traced back to 1990s, where neural algorithms for solving differential equations were proposed [4–8].”
Multi-Fidelity Physics-Constrained Neural Network and Its Application in Materials Modeling (2019, background): “the information of both boundary conditions and the model form [29,30].”