# Parameterized Physics-informed Neural Networks for Parameterized PDEs

arXiv: https://arxiv.org/abs/2408.09446 · HTML: https://mlsanity.com/paper/2408.09446/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 40 references that are in the map.
- First posted on arXiv 18 Aug 2024.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [An Overview of Multi-Task Learning in Deep Neural Networks](https://mlsanity.com/paper/1706.05098/) (2017)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition](https://mlsanity.com/paper/2211.08760/) (2022)

## Built on by (most cited first, 30 of 38)

- [FP64 is All You Need: Rethinking Failure Modes in Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.10949/) (2025)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [Sub-Sequential Physics-Informed Learning with State Space Model](https://mlsanity.com/paper/2502.00318/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solving](https://mlsanity.com/paper/2602.13616/) (2026)
- [Physics-informed reduced order model with conditional neural fields](https://mlsanity.com/paper/2412.05233/) (2024)
- [Governing Equation Discovery from Data Based on Differential Invariants](https://mlsanity.com/paper/2505.18798/) (2025)
- [Solving Differential Equations with Constrained Learning](https://mlsanity.com/paper/2410.22796/) (2024)
- [Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?](https://mlsanity.com/paper/2412.19235/) (2024)
- [Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networks](https://mlsanity.com/paper/2503.17402/) (2025)
- [Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning](https://mlsanity.com/paper/2509.24332/) (2025)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Unveiling the Potential of Superexpressive Networks in Implicit Neural Representations](https://mlsanity.com/paper/2503.21166/) (2025)
- [Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusion](https://mlsanity.com/paper/2603.12676/) (2026)
- [A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions](https://mlsanity.com/paper/2604.06001/) (2026)
- [Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.26999/) (2026)
- [Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving](https://mlsanity.com/paper/2606.23359/) (2026)
- [Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs](https://mlsanity.com/paper/2607.23466/) (2026)
- [MaD-Scientist: AI-based Scientist solving Convection-Diffusion-Reaction Equations Using Massive PINN-Based Prior Data](https://mlsanity.com/paper/2410.06442/) (2024)
- [Engineering application of physics-informed neural networks for Saint-Venant torsion](https://mlsanity.com/paper/2505.12389/) (2025)
- [Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systems](https://mlsanity.com/paper/2506.19609/) (2025)
- [IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation](https://mlsanity.com/paper/2509.07245/) (2025)
- [PIANO: Physics-informed Dual Neural Operator for Precipitation Nowcasting](https://mlsanity.com/paper/2512.01062/) (2025)
- [PGOT: A Physics-Geometry Operator Transformer for Complex PDEs](https://mlsanity.com/paper/2512.23192/) (2025)
- [Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridge](https://mlsanity.com/paper/2603.22319/) (2026)
- [Material-agnostic temperature field prediction for metal additive manufacturing via a parametric PINN framework](https://mlsanity.com/paper/2604.14562/) (2026)
- [Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes](https://mlsanity.com/paper/2604.24745/) (2026)

## Abstract (arXiv)

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P2INNs). P2INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P2INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known "failure modes".