The field of partial differential equations (PDEs) is experiencing a significant shift with the integration of neural networks, leading to innovative methods for solving complex problems. A key direction is the development of hybrid approaches that combine traditional numerical methods with neural networks, enabling the accurate capture of sharp discontinuities and maintaining temporal consistency. Another area of focus is the design of physics-informed neural networks that can effectively model dynamical systems, ensuring stability and accuracy in predictions. Noteworthy papers include: A Hybrid Discontinuous Galerkin Neural Network Method for Solving Hyperbolic Conservation Laws with Temporal Progressive Learning, which introduces a hybrid framework that couples discontinuous Galerkin discretizations with a temporally progressive neural network architecture. PIANO: Physics Informed Autoregressive Network, which redesigns PINNs to model dynamical systems autoregressively, achieving stability through autoregressive modeling.
Advances in Solving Partial Differential Equations with Neural Networks
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A Hybrid Discontinuous Galerkin Neural Network Method for Solving Hyperbolic Conservation Laws with Temporal Progressive Learning
PIANO: Physics Informed Autoregressive Network
A novel auxiliary equation neural networks method for exactly explicit solutions of nonlinear partial differential equations
An implicit-explicit BDF-Galerkin scheme of second order for the nonlinear thermistor problem
A convergence framework for energy minimisation of linear self-adjoint elliptic PDEs in nonlinear approximation spaces
Spectral-Prior Guided Multistage Physics-Informed Neural Networks for Highly Accurate PDE Solutions
Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models
Energy-Equidistributed Moving Sampling Physics-informed Neural Networks for Solving Conservative Partial Differential Equations
Automated Runge-Kutta-Nystr\"om time stepping for finite element methods in Irksome
D3PINNs: A Novel Physics-Informed Neural Network Framework for Staged Solving of Time-Dependent Partial Differential Equations
Tsunami modeling with dynamic seafloors: a high-order solver validated with shallow water benchmarks