# Solving Differential Equations Using Neural Network Solution Bundles

arXiv: https://arxiv.org/abs/2006.14372 · HTML: https://mlsanity.com/paper/2006.14372/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2020.

## Built on by (most cited first, 21 of 21)

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- [Uncertainty Quantification in Neural Differential Equations](https://mlsanity.com/paper/2111.04207/) (2021)
- [One-Shot Transfer Learning for Nonlinear ODEs](https://mlsanity.com/paper/2311.14931/) (2023)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
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- [Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media](https://mlsanity.com/paper/2305.06802/) (2023)
- [When and Why Adversarial Training Improves PINNs: A Neural Tangent Kernel Perspective](https://mlsanity.com/paper/2605.15959/) (2026)
- [Adversarial Sampling for Solving Differential Equations with Neural Networks](https://mlsanity.com/paper/2111.12024/) (2021)
- [Optimization-Informed Neural Networks](https://mlsanity.com/paper/2210.02113/) (2022)
- [Parallel-in-Time Solutions with Random Projection Neural Networks](https://mlsanity.com/paper/2408.09756/) (2024)
- [Learning embeddings of non-linear PDEs: the Burgers' equation](https://mlsanity.com/paper/2603.07812/) (2026)
- [Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations](https://mlsanity.com/paper/2605.01634/) (2026)
- [Recovering Sharp Conductivity Features in the Finite-Data Calder'on Problem with Physics-Informed Neural Networks](https://mlsanity.com/paper/2606.28158/) (2026)
- [Gravitational Duals from Equations of State II: Large Hierarchies and False Vacua](https://mlsanity.com/paper/2606.30117/) (2026)
- [Physics-Informed Neural Embeddings of PDE Solution Families](https://mlsanity.com/paper/2607.06348/) (2026)
- [Physics-Informed Neural Networks to Infer the Perpendicular Energy Conductivity in the Scrape-Off Layer of Stellarator Devices](https://mlsanity.com/paper/2609.11628/) (2026)

## Abstract (arXiv)

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.