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Multi-fidelity Learning of Reduced Order Models for Parabolic PDE Constrained Optimization

Benedikt Klein, Mario Ohlberger

TL;DR

This work tackles parameter optimization for parabolic PDEs by fusing certified reduced-order models with data-driven surrogates inside a hierarchical, trust-region framework. It introduces an on-the-fly multi-fidelity hierarchy (FOM, RB-ROM, ML surrogate) and harnesses an a posteriori error estimator to guide model selection and region-of-validation within a global optimization scheme. The key contributions include a posteriori error bounds for RB and ML surrogates, a dynamic space-adaptation strategy (RB enrichment) and a relaxed trust-region algorithm that allows uncertified ML models to participate in optimization while maintaining convergence. Numerical experiments on a building-temperature control problem show substantial speed-ups with kernel-based ML surrogates, though challenges remain in error certification and generalization; the results point to future work in physics-informed ML and refined error control to broaden practical impact.

Abstract

This article builds on the recently proposed RB-ML-ROM approach for parameterized parabolic PDEs and proposes a novel hierarchical Trust Region algorithm for solving parabolic PDE constrained optimization problems. Instead of using a traditional offline/online splitting approach for model order reduction, we adopt an active learning or enrichment strategy to construct a multi-fidelity hierarchy of reduced order models on-the-fly during the outer optimization loop. The multi-fidelity surrogate model consists of a full order model, a reduced order model and a machine learning model. The proposed hierarchical framework adaptively updates its hierarchy when querying parameters, utilizing a rigorous a posteriori error estimator in an error aware trust region framework. Numerical experiments are given to demonstrate the efficiency of the proposed approach.

Multi-fidelity Learning of Reduced Order Models for Parabolic PDE Constrained Optimization

TL;DR

This work tackles parameter optimization for parabolic PDEs by fusing certified reduced-order models with data-driven surrogates inside a hierarchical, trust-region framework. It introduces an on-the-fly multi-fidelity hierarchy (FOM, RB-ROM, ML surrogate) and harnesses an a posteriori error estimator to guide model selection and region-of-validation within a global optimization scheme. The key contributions include a posteriori error bounds for RB and ML surrogates, a dynamic space-adaptation strategy (RB enrichment) and a relaxed trust-region algorithm that allows uncertified ML models to participate in optimization while maintaining convergence. Numerical experiments on a building-temperature control problem show substantial speed-ups with kernel-based ML surrogates, though challenges remain in error certification and generalization; the results point to future work in physics-informed ML and refined error control to broaden practical impact.

Abstract

This article builds on the recently proposed RB-ML-ROM approach for parameterized parabolic PDEs and proposes a novel hierarchical Trust Region algorithm for solving parabolic PDE constrained optimization problems. Instead of using a traditional offline/online splitting approach for model order reduction, we adopt an active learning or enrichment strategy to construct a multi-fidelity hierarchy of reduced order models on-the-fly during the outer optimization loop. The multi-fidelity surrogate model consists of a full order model, a reduced order model and a machine learning model. The proposed hierarchical framework adaptively updates its hierarchy when querying parameters, utilizing a rigorous a posteriori error estimator in an error aware trust region framework. Numerical experiments are given to demonstrate the efficiency of the proposed approach.

Paper Structure

This paper contains 21 sections, 4 theorems, 82 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Adjoint Problem
  4. Reduced basis method
  5. A Posteriori Error Estimator
  6. Space Construction
  7. Machine learning surrogates
  8. Error Estimates for ML Surrogates
  9. General Considerations
  10. Trust region multi-fidelity optimization
  11. Backtracking Procedure
  12. ---optimization
  13. Numerical experiments
  14. Problem
  15. Kernel methods
  16. ...and 6 more sections

Key Result

Lemma 3.3

Let the trajectory $u \in Q^\text{pr}_{\Delta t}(0,T; V_\text{RB})$ be given and let ${p(u; \mu) \in Q^\text{ad}_{\Delta t}(0,T; V_\text{RB})}$ solve eq:dual_ROM_PDE for $\mu \in \mathcal{P}$ and the right-hand side defined by $u$. Furthermore, let $u_h(\mu) \in Q^\text{pr}_{\Delta t}(0,T; V_h)$ and and

Figures (4)

  • Figure 1: Schematic visualization of the abstract trust region algorithm.
  • Figure 2: Simplified flowchart illustrating the inner loop of the relaxed -- algorithm, as described in Algorithm \ref{['algo:TR_opt']}, after the warm-up phase.
  • Figure 3: Comparison of the desired state $g_{\text{ref}}$ (left), the final state obtained using the relaxed -- algorithm with initial guess $\mu^{(0)} = (0.026, 0.020)$ (middle), and their absolute differences (right) at the final time step ($K = 10000$).
  • Figure 4: Evaluation times of the primal models for each queried parameter during the run performed by the relaxed -- algorithm, starting at $\mu^{(0)} = (0.026, 0.020)$. The colors indicate the model used: (red), - (blue), and - (orange). Additionally, the scope of the -s $M^{(i)}$ is shown.

Theorems & Definitions (18)

  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2: Residual Operators
  • Lemma 3.3: Error Bounds
  • proof
  • Theorem 3.4: A posteriori objective error estimates
  • proof
  • Remark 3.5
  • Definition 4.1
  • ...and 8 more