# Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systems

arXiv: https://arxiv.org/abs/2512.11834 · HTML: https://mlsanity.com/paper/2512.11834/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 52 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## 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)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [Using machine learning to correct model error in data assimilation and forecast applications](https://mlsanity.com/paper/2010.12605/) (2020)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction](https://mlsanity.com/paper/1904.10904/) (2019)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Learning About Structural Errors in Models of Complex Dynamical Systems](https://mlsanity.com/paper/2401.00035/) (2023)
- [Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks](https://mlsanity.com/paper/2109.11354/) (2021)
- [Physics-guided Deep Markov Models for Learning Nonlinear Dynamical Systems with Uncertainty](https://mlsanity.com/paper/2110.08607/) (2021)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter](https://mlsanity.com/paper/2306.04315/) (2023)
- [Tangent: Automatic Differentiation Using Source Code Transformation in Python](https://mlsanity.com/paper/1711.02712/) (2017)
- [Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step case](https://mlsanity.com/paper/2503.11196/) (2025)

## Abstract (arXiv)

The accurate estimation of the state of complex uncertain physical systems requires reconciling theoretical models, with inherent imperfections, with noisy experimental data. In this work, we propose an effective hybrid approach that combines physics-based modeling with data-driven learning to enhance state estimation and further prediction. Our method builds upon the Parameterized Background Data-Weak (PBDW) framework, which naturally integrates a reduced-order representation of the best-available model with measurement data to account for both anticipated and unanticipated uncertainties. To address model discrepancies not captured by the reduced-order space, and learn the structure of model deviation, we incorporate a Deep Operator Network (DeepONet) constrained to be an orthogonal complement of the best-knowledge manifold. This ensures that the learned correction targets only the unknown components of model bias, preserving the interpretability and fidelity of the physical model. An optimal sensor placement strategy is also investigated to maximize information gained from measurements. We validate the proposed approach on a representative problem involving the Helmholtz equation under various sources of modeling error, including those arising from boundary conditions and source terms.