# Active Learning of Model Discrepancy with Bayesian Experimental Design

arXiv: https://arxiv.org/abs/2502.05372 · HTML: https://mlsanity.com/paper/2502.05372/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 70 references that are in the map.
- First posted on arXiv 7 Feb 2025.

## Builds on (most cited first, 25 of 25)

- [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)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Deep Adaptive Design: Amortizing Sequential Bayesian Experimental Design](https://mlsanity.com/paper/2103.02438/) (2021)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Implicit Deep Adaptive Design: Policy-Based Experimental Design without Likelihoods](https://mlsanity.com/paper/2111.02329/) (2021)
- [Optimal experimental design: Formulations and computations](https://mlsanity.com/paper/2407.16212/) (2024)
- [Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks](https://mlsanity.com/paper/1808.03620/) (2018)
- [Sequential Bayesian optimal experimental design via approximate dynamic programming](https://mlsanity.com/paper/1604.08320/) (2016)
- [Bayesian Sequential Optimal Experimental Design for Nonlinear Models Using Policy Gradient Reinforcement Learning](https://mlsanity.com/paper/2110.15335/) (2021)
- [Sequential Bayesian Experimental Design for Implicit Models via Mutual Information](https://mlsanity.com/paper/2003.09379/) (2020)
- [Large-scale Bayesian optimal experimental design with derivative-informed projected neural network](https://mlsanity.com/paper/2201.07925/) (2022)
- [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Learning About Structural Errors in Models of Complex Dynamical Systems](https://mlsanity.com/paper/2401.00035/) (2023)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization](https://mlsanity.com/paper/2208.09104/) (2022)
- [Bayesian Optimal Experimental Design for Inferring Causal Structure](https://mlsanity.com/paper/2103.15229/) (2021)
- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [Metrics for Bayesian Optimal Experiment Design under Model Misspecification](https://mlsanity.com/paper/2304.07949/) (2023)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [Sequential Bayesian experimental design for calibration of expensive simulation models](https://mlsanity.com/paper/2305.16506/) (2023)
- [Measure This, Not That: Optimizing the Cost and Model-Based Information Content of Measurements](https://mlsanity.com/paper/2406.09557/) (2024)

## Built on by (most cited first, 5 of 5)

- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach](https://mlsanity.com/paper/2504.20319/) (2025)
- [Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modeling](https://mlsanity.com/paper/2602.17089/) (2026)
- [Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance](https://mlsanity.com/paper/2601.16425/) (2026)

## Abstract (arXiv)

Digital twins have been actively explored in many engineering applications, such as manufacturing and autonomous systems. However, model discrepancy is ubiquitous in most digital twin models and has significant impacts on the performance of using those models. In recent years, data-driven modeling techniques have been demonstrated promising in characterizing the model discrepancy in existing models, while the training data for the learning of model discrepancy is often obtained in an empirical way and an active approach of gathering informative data can potentially benefit the learning of model discrepancy. On the other hand, Bayesian experimental design (BED) provides a systematic approach to gathering the most informative data, but its performance is often negatively impacted by the model discrepancy. In this work, we build on sequential BED and propose an efficient approach to iteratively learn the model discrepancy based on the data from the BED. The performance of the proposed method is validated by a classical numerical example governed by a convection-diffusion equation, for which full BED is still feasible. The proposed method is then further studied in the same numerical example with a high-dimensional model discrepancy, which serves as a demonstration for the scenarios where full BED is not practical anymore. An ensemble-based approximation of information gain is further utilized to assess the data informativeness and to enhance learning model discrepancy. The results show that the proposed method is efficient and robust to the active learning of high-dimensional model discrepancy, using data suggested by the sequential BED. We also demonstrate that the proposed method is compatible with both classical numerical solvers and modern auto-differentiable solvers.