# Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations

arXiv: https://arxiv.org/abs/2605.01634 · HTML: https://mlsanity.com/paper/2605.01634/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 17 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Solving Differential Equations Using Neural Network Solution Bundles](https://mlsanity.com/paper/2006.14372/) (2020)
- [One-Shot Transfer Learning for Nonlinear ODEs](https://mlsanity.com/paper/2311.14931/) (2023)
- [One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs](https://mlsanity.com/paper/2511.11137/) (2025)
- [PTL-PINNs: Perturbation-Guided Transfer Learning with Physics- Informed Neural Networks for Nonlinear Systems](https://mlsanity.com/paper/2601.12093/) (2026)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by embedding governing laws into the training objective. A persistent limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, boundary/initial condition, or parameter setting. One-shot transfer learning (OTL) addresses this bottleneck for linear operators by freezing a pretrained latent representation and computing optimal output weights in closed form, but for nonlinear problems closed-form adaptation is generally unavailable because the loss is nonconvex in the output layer. In this paper we substantially broaden the class of nonlinearities amenable to one-shot PINN transfer by combining OTL with Chebyshev polynomial surrogates. We approximate general smooth weakly nonlinear terms by truncated Chebyshev expansions over a prescribed solution range, yielding a polynomial nonlinearity that can be handled by a perturbative decomposition into linear subproblems. A multi-head PINN learns a reusable latent space associated with the dominant linear operator; at test time, solutions to new instances are obtained via a sequence of closed-form linear solves in the output layer, without retraining the network body. We provide a unified derivation of the framework for ODEs and PDEs and demonstrate accuracy and fast online adaptation on nonlinear benchmarks, including non-polynomial and singular ODE nonlinearities as well as a reaction-diffusion PDE with saturating kinetics, demonstrating the method's utility in many-query regimes.