Neural-network-based approximations for solving partial differential equations

By M. Dissanayake, N. Phan-Thien. 1994

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Citations in the corpus

PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs (2019, background): “This leads to a physics-informed neural network (PINN), where physical conservation laws and prior physical knowledge are encoded into the neural networks [1,12].”
Efficient training of physics‐informed neural networks via importance sampling (2021, background): “The idea of using physics-informed training for neural network solutions of differential equations was first introduced in [1, 5, 6], where neural networks solutions for initial/boundary value problems were developed.”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, methodology): “A common technique is to use an ML model as a surrogate for the solution to reduce computation time [65, 140].”
Weak Adversarial Networks for High-dimensional Partial Differential Equations (2019, unknown): “In contrast to decomposing the solution into two parts, the idea of approximating the solution of PDEs by a single neural network is considered in [26], which is not capable of dealing with high-dimensional problems.”
Characterizing possible failure modes in physics-informed neural networks (2021, background): “This idea has been introduced in different forms in the past (for instance [5, 18, 25, 28, 32]).”
DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, methodology): “Alternatively, one could use the PDE in strong form directly [15, 52, 30, 31, 7, 50, 47]; in this form, automatic”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, methodology): “A common technique is to use an ML model as a surrogate for the solution to reduce computation time [75, 159].”
Graph Element Networks: adaptive, structured computation and memory (2019, methodology): “Although GENs were originally inspired by ﬁnite element methods that model PDEs, we show that they are much more widely applicable.”
Multi-Fidelity Physics-Constrained Neural Network and Its Application in Materials Modeling (2019, background): “By transforming the original PDEs into their weighted residual forms, the prior knowledge of model forms and boundary values can be incorporated as penalty functions during the training of ANNs [25].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, background): “The brilliance of the first PINN articles (Raissi et al, 2017c,d) lies in resurrecting the concept of optimizing a problem with a physical constraint by approximating the unknown function with a neural network (Dissanayake and Phan-Thien, 1994) and then extending this concept to a hybrid data-equati”