# A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries

arXiv: https://arxiv.org/abs/2504.02459 · HTML: https://mlsanity.com/paper/2504.02459/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 91 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Builds on (most cited first, 30 of 46)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [COIN++: Neural Compression Across Modalities](https://mlsanity.com/paper/2201.12904/) (2022)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Advances and Challenges in Meta-Learning: A Technical Review](https://mlsanity.com/paper/2307.04722/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Simulation of hyperelastic materials in real-time using Deep Learning](https://mlsanity.com/paper/1904.06197/) (2019)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systems](https://mlsanity.com/paper/2503.13695/) (2025)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Aero-Nef: Neural Fields for Rapid Aircraft Aerodynamics Simulations](https://mlsanity.com/paper/2407.19916/) (2024)
- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [U-net architectures for fast prediction of incompressible laminar flows](https://mlsanity.com/paper/1910.13532/) (2019)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [Time Series Continuous Modeling for Imputation and Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2306.05880/) (2023)

## Built on by (most cited first, 5 of 5)

- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [Architecture Shapes Transfer Specificity in Implicit Neural Representations](https://mlsanity.com/paper/2606.06827/) (2026)
- [Variational Parameter Calibration with Physics-Aware Latent-Space Surrogates](https://mlsanity.com/paper/2608.11435/) (2026)
- [Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage](https://mlsanity.com/paper/2609.02194/) (2026)

## Abstract (arXiv)

In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics-informed encoder-decoder network to establish the mapping between continuous parameter and solution spaces. The decoder constructs the parametric solution field by leveraging an implicit neural field network conditioned on a latent or feature code. Instance-specific codes are derived through a PDE encoding process based on the second-order meta-learning technique. In training and inference, a physics-informed loss function is minimized during the PDE encoding and decoding. iFOL expresses the loss function in an energy or weighted residual form and evaluates it using discrete residuals derived from standard numerical PDE methods. This approach results in the backpropagation of discrete residuals during both training and inference. iFOL features several key properties: (1) its unique loss formulation eliminates the need for the conventional encode-process-decode pipeline previously used in operator learning with conditional neural fields for PDEs; (2) it not only provides accurate parametric and continuous fields but also delivers solution-to-parameter gradients without requiring additional loss terms or sensitivity analysis; (3) it can effectively capture sharp discontinuities in the solution; and (4) it removes constraints on the geometry and mesh, making it applicable to arbitrary geometries and spatial sampling (zero-shot super-resolution capability). We critically assess these features and analyze the network's ability to generalize to unseen samples across both stationary and transient PDEs. The overall performance of the proposed method is promising, demonstrating its applicability to a range of challenging problems in computational mechanics.