Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems
This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction-diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction-diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.
Code (0)
등록된 구현이 없습니다.
Tasks
Inference OptimizationMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Predictor-Corrector Enhanced Transformers with Exponential Moving Average Coefficient Learning
Residual networks, as discrete approximations of Ordinary Differential Equations (ODEs), have inspired significant advancements in neural network design, including multistep methods, high-order methods, and multi-particl…
Abstractive Text SummarizationLanguage ModelingLanguage ModellingMachine Translation+2PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting
Coupled spatiotemporal forecasting is important for predicting the future evolution of multiple interacting dynamical systems, such as in climate models. However, existing methods are severely constrained by the persiste…
Causality-Inspired Safe Residual Correction for Multivariate Time Series
While modern multivariate forecasters such as Transformers and GNNs achieve strong benchmark performance, they often suffer from systematic errors at specific variables or horizons and, critically, lack guarantees agains…
Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parame…
Continuous-Time Probabilistic Correctors for Uncertainty-Aware Physics-Based Spacecraft Trajectory Forecasting
Long-horizon spacecraft trajectory forecasting suffers from error accumulation due to the absence of corrective observations in the forecast regime, making reliable uncertainty estimation crucial for safety-critical deci…
Trajectory Forecasting