# Active operator learning with predictive uncertainty quantification for partial differential equations arXiv: https://arxiv.org/abs/2503.03178 ยท HTML: https://mlsanity.com/paper/2503.03178/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Builds on 38 of its 66 references that are in the map. - First posted on arXiv 5 Mar 2025. ## Builds on (most cited first, 30 of 38) - [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019) - [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015) - [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016) - [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020) - [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019) - [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017) - [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020) - [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021) - [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024) - [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021) - [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022) - [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021) - [A Deeper Look into Aleatoric and Epistemic Uncertainty Disentanglement](https://mlsanity.com/paper/2204.09308/) (2022) - [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022) - [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022) - [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023) - [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023) - [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024) - [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022) - [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021) - [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024) - [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022) - [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022) - [Accelerated Linearized Laplace Approximation for Bayesian Deep Learning](https://mlsanity.com/paper/2210.12642/) (2022) - [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024) - [PPDONet: Deep Operator Networks for Fast Prediction of Steady-State Solutions in Disk-Planet Systems](https://mlsanity.com/paper/2305.11111/) (2023) - [Variational Linearized Laplace Approximation for Bayesian Deep Learning](https://mlsanity.com/paper/2302.12565/) (2023) - [D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators](https://mlsanity.com/paper/2310.18888/) (2023) ## Built on by (most cited first, 12 of 12) - [PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations](https://mlsanity.com/paper/2512.23056/) (2025) - [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025) - [Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networks](https://mlsanity.com/paper/2509.00663/) (2025) - [Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition](https://mlsanity.com/paper/2602.15632/) (2026) - [Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations](https://mlsanity.com/paper/2605.17107/) (2026) - [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026) - [Energy-efficient operation of neural operators for virtual sensing](https://mlsanity.com/paper/2609.30580/) (2026) - [Operator Learning for Consolidation: An Architectural Comparison for DeepONet Variants](https://mlsanity.com/paper/2507.10368/) (2025) - [GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries](https://mlsanity.com/paper/2509.24117/) (2025) - [Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization](https://mlsanity.com/paper/2605.15806/) (2026) - [Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields](https://mlsanity.com/paper/2608.09071/) (2026) - [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026) ## Abstract (arXiv) With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. We propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework's uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.