Inverse Problems & Data Assimilation
Model-data integration for PDE systems through continuous assimilation, ensemble filtering, inverse reconstruction, and uncertainty-aware computation.
Overview
Predictive PDE models are often limited by uncertain parameters, incomplete initial data, sparse observations, and model discrepancy. My research develops mathematical and computational methods that combine observations with PDE models to improve state estimation, parameter reconstruction, and predictive accuracy.
The work spans deterministic continuous data assimilation, ensemble-based filtering, score-based inference, and inverse PDE algorithms that integrate machine learning with finite element forward models. A common theme is to retain the structure and reliability of the underlying numerical solver while incorporating data efficiently.
Continuous Data Assimilation
Continuous data assimilation (CDA) incorporates observational information directly into the evolution equations through a feedback or nudging mechanism. For Biot's poroelasticity system, my work develops a CDA formulation in which sparse pressure observations are used to drive the computed state toward the reference solution.
- Convergence analysis. Establish conditions under which the assimilated solution approaches the reference state.
- Sparse observations. Incorporate measurements through interpolation operators at resolutions different from the computational mesh.
- Coupled physics. Combine nudging with iterative fixed-stress splitting for pressure-displacement systems.
- Heterogeneity. Test the method for heterogeneous and nonlinear permeability settings.
Ensemble and Score-Based Filtering
For nonlinear and uncertain dynamical systems, ensemble methods provide a flexible way to represent evolving uncertainty. Recent work applies the Ensemble Score Filter (EnSF) to two-phase flow in porous media, where uncertain permeability and incomplete measurements make state estimation challenging.
The EnSF framework uses a score-based representation of the filtering distribution together with an analytical update, providing an alternative to conventional particle and ensemble filters for high-dimensional nonlinear problems.
- state estimation for pressure, velocity, and saturation;
- uncertain heterogeneous permeability;
- incomplete observational data;
- high-dimensional nonlinear filtering without training a separate neural score model.
Scalable Ensemble Kalman Methods
Another direction focuses on the computational structure of ensemble Kalman filtering. A conjugate-gradient formulation of the EnKF analysis step replaces a direct linear solve with an iterative formulation and exposes opportunities for dimension reduction and scalable computation. This line of work is motivated by the cost of repeated assimilation in large PDE systems.
Inverse PDE Problems
Inverse PDE problems seek unknown coefficients, sources, or physical properties from indirect measurements. Recent work combines finite element simulation with machine learning for inverse elliptic problems using the Dirichlet-to-Neumann map. The forward PDE remains embedded in the learning loop, providing a physics-based connection between boundary data and the inferred interior coefficient.
- Boundary observations. Reconstruct interior properties from Dirichlet-to-Neumann data.
- Partial and noisy data. Design the inverse procedure for incomplete measurements.
- Discontinuous coefficients. Use loss constructions suited to non-smooth material fields.
- Hybrid computation. Retain a finite element forward solver inside the inverse-learning framework.
Uncertainty-Aware Inference
A broader objective is to distinguish and quantify different sources of uncertainty in model-data integration: observational uncertainty, parameter uncertainty, model discrepancy, and numerical approximation error. This distinction is important when an inferred parameter or state is subsequently used for prediction, optimization, or decision-making.
Current Directions
- High-dimensional PDE inference. Scalable filtering and inverse methods for large state and parameter spaces.
- Bayesian and uncertainty-aware methods. Posterior characterization and uncertainty propagation for PDE inverse problems.
- Measurement design. Determine which observations are most informative for the quantities or parameters of interest.
- Learned surrogates for inference. Use operator learning and reduced models while controlling the effect of surrogate error on the inverse solution.
Selected Publications
- A Conjugate Gradient Formulation of the EnKF Algorithm. In revision, Computers & Mathematics with Applications, 2026. arXiv
- A Machine Learning and Finite Element Framework for Inverse Elliptic PDEs via Dirichlet-to-Neumann Mapping. Journal of Computational and Applied Mathematics, 486, 117679, 2026. arXiv Journal
- Ensemble Score Filter for Data Assimilation of Two-Phase Flow Models in Porous Media. Journal of Computational Physics, 544, 114416, 2026. arXiv Journal
- Convergence and Numerical Simulations of the Continuous Data Assimilation for Biot's Poroelasticity System. Computers & Mathematics with Applications, 202, 20-37, 2026. Journal
Related Funded Projects
- DOE ASCR: From Uncertainty to Trust: A Scalable Stochastic Computing Framework for Uncertainty Reduction in Artificial Intelligence.
- NSF REU Site: Applied Mathematics in Real World Problems — project on data assimilation and parameter prediction for subsurface flows and simplified fluid models.
- DOE Science Foundations for the Energy Earthshots: Advanced Multi-Physics Machine Learning for Subsurface Energy Systems Across Scales.