Solving Differential Equations Using Neural Network Solution Bundles
read the original abstract
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.
This paper has not been read by Pith yet.
Forward citations
Cited by 4 Pith papers
-
Recovering Sharp Conductivity Features in the Finite-Data Calder\'on Problem with Physics-Informed Neural Networks
A PINN framework with separate networks for conductivity and potentials, multiscale wavelet excitations, and FFE recovers dominant conductivity structures from finite DtN data with 3-12% relative error on synthetic te...
-
When and Why Adversarial Training Improves PINNs: A Neural Tangent Kernel Perspective
Adversarial training improves PINNs by using the discriminator to mitigate spectral bias and stiffness, with a new NTK-based framework providing theoretical grounding and a practical algorithm.
-
Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations
Chebyshev polynomial surrogates enable one-shot closed-form adaptation of PINNs for a broader class of nonlinear ODEs and PDEs by decomposing them into linear subproblems.
-
Gravitational Duals from Equations of State II: Large Hierarchies and False Vacua
Extends PINN-based holographic inverse problem solving to false vacuum and large hierarchy regimes, claiming accurate scalar potential reconstruction from thermodynamic data despite numerical stiffness.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.