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A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs

arXiv 2110.13361 ↗First posted 26 Oct 2021physics.comp-ph cs.LGcited by 32

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Abstract (arXiv)

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) - and, in fact, machine learning (ML) in general. This paper is meant to move towards addressing the latter through the study of PINNs on new tasks, for which parameterized PDEs provides a good testbed application as tasks can be easily defined in this context. Following the ML world, we introduce metalearning of PINNs with application to parameterized PDEs. By introducing metalearning and transfer learning concepts, we can greatly accelerate the PINNs optimization process. We present a survey of model-agnostic metalearning, and then discuss our model-aware metalearning applied to PINNs as well as implementation considerations and algorithmic complexity. We then test our approach on various canonical forward parameterized PDEs that have been presented in the emerging PINNs literature.

Key facts. Cited by 32 papers in the map (data through 14 Oct 2026). Builds on 37 of its 80 references that are in the map. First posted on arXiv 26 Oct 2021.

Builds on

37 of 80 references.

References · 37

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25 more
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  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 2; 2023: 4; 2024: 8; 2025: 11; 2026: 7.

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  1. 2024Generating synthetic data for neural operatorscited by 11
  2. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 11
  3. 2022Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problemscited by 9
  4. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  5. 2025Towards a Physics Foundation Modelcited by 8
  6. 2022Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networkscited by 6
  7. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  8. 2023Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effectcited by 5
  9. 2024Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networkscited by 5
  10. 2025MetaSym: A Symplectic Meta-learning Framework for Physical Intelligencecited by 5
  11. 2025Adaptive Physics-informed Neural Networks: A Surveycited by 5
  12. 2026Out-of-Distribution Generalization for Neural Physics Solverscited by 5
20 more
  1. 2023Controlled physics-informed data generation for deep learning-based remaining useful life prediction under unseen operation conditionscited by 4
  2. 2024Resolution invariant deep operator network for PDEs with complex geometriescited by 4
  3. 2024Learning solutions of parametric Navier-Stokes with physics-informed neural networkscited by 4
  4. 2023iPINNs: Incremental learning for Physics-informed neural networkscited by 3
  5. 2023Branched Latent Neural Mapscited by 3
  6. 2024Common pitfalls to avoid while using multiobjective optimization in machine learningcited by 2
  7. 2024Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networkscited by 2
  8. 2024Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metricscited by 1
  9. 2024Towards Foundation Model for Chemical Reactor Modeling: Meta-Learning with Physics-Informed Adaptationcited by 1
  10. 2025Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEscited by 1
  11. 2026Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Familiescited by 1
  12. 2026Equation Recast for Canonical Operator Learning Across Parametric PDEscited by 1
  13. 2025Multi-level datasets training method in Physics-Informed Neural Networkscited by 0
  14. 2025Learning and Transferring Physical Models through Derivativescited by 0
  15. 2025Latent Representation Learning of Multi-scale Thermophysics: Application to Dynamics in Shocked Porous Energetic Materialcited by 0
  16. 2025LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEscited by 0
  17. 2025SUPN: Shallow Universal Polynomial Networks for Efficient Function Approximationcited by 0
  18. 2026Meta-Learned Basis Adaptation for Parametric Linear PDEscited by 0
  19. 2026Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equationscited by 0
  20. 2026Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricingcited by 0