# Adapting neural operators for mechanics decisions under changing operating conditions

arXiv: https://arxiv.org/abs/2609.33978 · HTML: https://mlsanity.com/paper/2609.33978/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [A digital twin framework for civil engineering structures](https://mlsanity.com/paper/2308.01445/) (2023)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)
- [Goal-Oriented A-Posteriori Estimation of Model Error as an Aid to Parameter Estimation](https://mlsanity.com/paper/2205.15131/) (2022)
- [From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing](https://mlsanity.com/paper/2503.05598/) (2025)
- [Model-Informed Joint Material-Structural Optimization of Hard-Magnetic Soft Materials](https://mlsanity.com/paper/2607.14397/) (2026)

## Abstract (arXiv)

Neural operators can accelerate repeated nonlinear mechanics calculations, but their accuracy can deteriorate as operating conditions move beyond the training range. This work studies whether high-fidelity solutions acquired during use can be reused to adapt a neural operator and improve subsequent mechanics-based command selection. Two hard-magnetic soft-material systems are simulated using high-fidelity finite-element (FE) models, providing reference solutions for evaluating surrogate predictions and selected commands. A neural operator predicts deformation from known material, loading, and magnetic-field inputs, while an empirical error estimator determines which predictions may be used for command selection. Selected FE evaluations supplement these predictions, and their complete loading paths are retained for periodic updates of the neural operator and estimator. In both examples, the fixed operator loses substantial accuracy when stiffness and loading move outside the training range. Updates using 16 acquired paths recover much of the lost accuracy while preserving accuracy in the nominal regime. Under the same FE evaluation budget, the updated operators also improve command selection, although the benefit varies with the operating condition. Error estimation is less consistent, with inaccurate predictions sometimes accepted and accurate predictions rejected. These results demonstrate that reusing high-fidelity loading paths can extend the useful operating range of a neural operator. However, improved forward accuracy alone does not guarantee reliable prediction acceptance, highlighting prediction-specific error assessment as a separate requirement for trustworthy decision making.