Neural algorithm for solving differential equations

1990

Finite difference equations are considered to solve differential equations numerically by utilizing minimization algorithms. Neural minimization algorithms for solving the finite difference equations are presented. Results of numerical simulation are described to demonstrate the method. Methods of implementing the algorithms are discussed. General features of the neural algorithms are discussed.

Citations in the corpus

• Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, background): “Lee and Kang (1990) developed highly parallel algorithms, based on neural networks, for solving finite difference equations (which are themselves approximations of original differential equations).”
• Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “In fact, the idea of using neural networks to solve PDEs is not new and can date back to the last century [11–14].”
• 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.”
• Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “…universal approximation achievements of the late 1980s, (Hornik et al, 1989); then in the early 90s several methodologies were proposed to use neural networks to approximate PDEs, like the work on constrained neural networks (Psichogios and Ungar, 1992; Lagaris et al, 1998) or (Lee and Kang, 1990).”
• Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, unknown): “Solve PDEs [147] [65] [140] [9] [215] [42] [264] [106] [235] [107] [131] [209] [234] [279] [281] [60] [177] [208] [231] [301] [71] [89] [129] [268] [194] [176] [16] [48] [218] [44] [60] [172] [235] [19] [51] [131] [76] [278] [194] [176] [206] [166]”
• 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].”
• Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, methodology): “, we refer to [328]–[335] and references cited therein for speeding up numerical solvers for realtime applications), and can be used in the context of digital twin when the same process is to be monitored time and again.”
• A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations (2018, background): “For instance, in [12–15], the computation of difference equations that appear in the finite difference solution for a PDE was accelerated by neural networks.”
• Implicit Neural Representations with Periodic Activation Functions (2020, background): “Neural networks have long been investigated in the context of solving differential equations (DEs) [37], and have previously been introduced as implicit representations for this task [38].”
• PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, background): “The idea of using neural networks to solve differential equations has been proposed decades ago [16–18] and recently experienced a resurgence of interest due to remarkable advances in deep learning [2, 19].”
• Weak Adversarial Networks for High-dimensional Partial Differential Equations (2019, methodology): “In [16], parallel neural networks are used to improve the efficiency of the finite difference method.”