[Paper Review] Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models

Sheroze Sheriffdeen, Jean C. Ragusa+3 more|arXiv (Cornell University)|Dec 17, 2019

Model Reduction and Neural Networks22 references16 citations

TL;DR

This paper proposes a deep learning-enhanced reduced order model (DL-ROM) that corrects projection-based ROMs by learning their error relative to high-fidelity PDE solutions, significantly improving accuracy while preserving physical consistency. The method accelerates PDE-constrained inverse problems, achieving near-high-fidelity accuracy with orders-of-magnitude faster evaluation, as demonstrated on elliptic PDEs and neutron transport problems with relative errors reduced to 0.01%.

ABSTRACT

Inverse problems are pervasive mathematical methods in inferring knowledge from observational and experimental data by leveraging simulations and models. Unlike direct inference methods, inverse problem approaches typically require many forward model solves usually governed by Partial Differential Equations (PDEs). This a crucial bottleneck in determining the feasibility of such methods. While machine learning (ML) methods, such as deep neural networks (DNNs), can be employed to learn nonlinear forward models, designing a network architecture that preserves accuracy while generalizing to new parameter regimes is a daunting task. Furthermore, due to the computation-expensive nature of forward models, state-of-the-art black-box ML methods would require an unrealistic amount of work in order to obtain an accurate surrogate model. On the other hand, standard Reduced-Order Models (ROMs) accurately capture supposedly important physics of the forward model in the reduced subspaces, but otherwise could be inaccurate elsewhere. In this paper, we propose to enlarge the validity of ROMs and hence improve the accuracy outside the reduced subspaces by incorporating a data-driven ML technique. In particular, we focus on a goal-oriented approach that substantially improves the accuracy of reduced models by learning the error between the forward model and the ROM outputs. Once an ML-enhanced ROM is constructed it can accelerate the performance of solving many-query problems in parametrized forward and inverse problems. Numerical results for inverse problems governed by elliptic PDEs and parametrized neutron transport equations will be presented to support our approach.

Motivation & Objective

• To address the computational bottleneck in PDE-constrained inverse problems, which require repeated high-fidelity forward solves.
• To improve the accuracy of reduced order models (ROMs) beyond their intrinsic limits without sacrificing physical consistency.
• To combine physics-preserving ROMs with data-driven deep learning to correct ROM errors using high-fidelity simulation and observational data.
• To enable efficient many-query inverse problems, such as parameter estimation and Bayesian inference, by accelerating the parameter-to-observable map.
• To decouple physical fidelity (from ROM) and accuracy enhancement (from deep learning), enabling robust and generalizable surrogate modeling.

Proposed method

• Construct a projection-based reduced-order model (ROM) using a reduced basis Φ to project the full-order model (FOM) onto a lower-dimensional subspace, ensuring physical consistency and computational efficiency.
• Train a deep neural network (DNN) to learn the discrepancy between the ROM solution and the high-fidelity FOM solution across a range of parameter realizations.
• Use a constrained least-squares approach to ensure the ROM preserves key physical properties such as conservation and stability.
• Train the DNN on a dataset of parameter inputs, corresponding high-fidelity FOM outputs, and ROM outputs to learn the error function e(u) = FOM(u) - ROM(u).
• Apply the trained DNN discrepancy function to correct ROM predictions, yielding a DL-enhanced ROM: FOM ≈ ROM + DNN(e(u)).
• Use the DL-enhanced ROM in place of the FOM for many-query problems, including inverse problems, to drastically reduce computational cost while maintaining high accuracy.

Experimental results

Research questions

• RQ1 Can a deep learning model effectively correct the error of a physics-preserving reduced order model to achieve high accuracy across diverse parameter regimes?
• RQ2 How does the DL-enhanced ROM compare in accuracy and efficiency to both standard ROMs and black-box deep learning surrogates in PDE-constrained inverse problems?
• RQ3 To what extent can the DL-enhanced ROM reduce the relative error in quantities of interest compared to high-fidelity solutions in complex benchmarks like neutron transport?
• RQ4 Does the decoupling of physics embedding (via ROM) and accuracy enhancement (via DNN) lead to more reliable and generalizable models than end-to-end deep learning?
• RQ5 Can the DL-enhanced ROM be effectively used in inverse problems requiring many forward evaluations, such as parameter estimation or Bayesian inference?

Key findings

• The DL-enhanced ROM reduced the relative quantity of interest error to 0.010% on a 22-dimensional ROM, compared to 1.08% for the high-fidelity diffusion FOM and 114.05% for a 3-dimensional ROM.
• Even with only 3 reduced basis vectors, the DL-enhanced ROM achieved a relative error of 0.86%, significantly outperforming the 114.05% error of the uncorrected ROM.
• The method achieved near-high-fidelity accuracy (0.01% error) while reducing computational cost by orders of magnitude compared to full-order models.
• The discrepancy function learned by the DNN effectively corrected systematic errors in the diffusion ROM, even when the ROM itself was limited by the underlying physics approximation.
• The approach outperformed the high-fidelity diffusion FOM (1.06% error) by learning the gap between diffusion and transport solutions, demonstrating the value of error modeling over direct approximation.
• The DL-enhanced ROM enabled efficient solution of a deterministic inverse problem for thermal conductivity reconstruction using sparse temperature observations, validating its utility in practical inverse settings.