# Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equations

arXiv: https://arxiv.org/abs/2510.10350 · HTML: https://mlsanity.com/paper/2510.10350/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 11 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
- [Quantifying Training Difficulty and Accelerating Convergence in Neural Network-Based PDE Solvers](https://mlsanity.com/paper/2410.06308/) (2024)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)

## Abstract (arXiv)

Operator learning provides a data-driven approach to approximating solution operators of partial differential equations, but its effectiveness depends strongly on how input and output functions are represented. Point-value representations can make the trainable map mesh-dependent and high-dimensional; snapshot-based POD/PCA reductions require aligned data and basis construction, while learned representations introduce additional trainable encoders, decoders, or neural bases. We propose the Fixed-Basis Coefficient-to-Coefficient Network (FB-C2CNet), which learns PDE solution maps in fixed, data-independent approximation spaces using prescribed bases as function encoders and decoders. Input observations are encoded by regularized least-squares projection onto bases such as finite element, random-feature, or radial-basis-function bases. A neural network maps the resulting input coefficients to output coefficients, and the fixed decoder reconstructs the solution at arbitrary target locations. This separation of basis selection from network training avoids neural basis learning and snapshot-based basis extraction, reduces the dimension of the trainable map, and lowers training cost. With suitable prescribed bases, FB-C2CNet also accommodates scattered, non-aligned, and sample-dependent observations. We analyze the stability--bias trade-off of regularized coefficient encoding and the intrinsic projection error determined by the output space. Experiments on elliptic, nonlinear time-dependent, weak-solution, high-dimensional, and inverse Stokes boundary-recovery problems demonstrate competitive accuracy with reduced trainable dimension and training cost, including for high-resolution and irregularly sampled data.