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IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation

arXiv 2509.07245 ↗First posted 8 Sep 2025cs.LGcited by 0

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Abstract (arXiv)

Solving inverse problems with Physics-Informed Neural Networks (PINNs) is computationally expensive for multi-query scenarios, as each new set of observed data requires a new, expensive training procedure. We present Inverse-Parameter Basis PINNs (IP-Basis PINNs), a meta-learning framework that extends the foundational work of Desai et al. (2022) to enable rapid and efficient inference for inverse problems. Our method employs an offline-online decomposition: a deep network is first trained offline to produce a rich set of basis functions that span the solution space of a parametric differential equation. For each new inverse problem online, this network is frozen, and solutions and parameters are inferred by training only a lightweight linear output layer against observed data. Key innovations that make our approach effective for inverse problems include: (1) a novel online loss formulation for simultaneous solution reconstruction and parameter identification, (2) a significant reduction in computational overhead via forward-mode automatic differentiation for PDE loss evaluation, and (3) a non-trivial validation and early-stopping mechanism for robust offline training. We demonstrate the efficacy of IP-Basis PINNs on three diverse benchmarks, including an extension to universal PINNs for unknown functional terms-showing consistent performance across constant and functional parameter estimation, a significant speedup per query over standard PINNs, and robust operation with scarce and noisy data.

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References · 20

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  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  5. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
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  8. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  9. 2021One-Shot Transfer Learning of Physics-Informed Neural Networkscited by 49
  10. 2022Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarioscited by 46
  11. 2024Parameterized Physics-informed Neural Networks for Parameterized PDEscited by 38
  12. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
8 more
  1. 2023GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEscited by 20
  2. 2021How to Avoid Trivial Solutions in Physics-Informed Neural Networkscited by 15
  3. 2025Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptationcited by 15
  4. 2022SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decompositioncited by 13
  5. 2024Second-Order Forward-Mode Automatic Differentiation for Optimizationcited by 5
  6. 2023One-Shot Transfer Learning for Nonlinear ODEscited by 3
  7. 2024Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networkscited by 2
  8. 2024Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysiscited by 1

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