# Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2604.26999 · HTML: https://mlsanity.com/paper/2604.26999/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 39 references that are in the map.
- First posted on arXiv 29 Apr 2026.

## Builds on (most cited first, 27 of 27)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [TADAM: Task dependent adaptive metric for improved few-shot learning](https://mlsanity.com/paper/1805.10123/) (2018)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Fast and Flexible Multi-Task Classification Using Conditional Neural Adaptive Processes](https://mlsanity.com/paper/1906.07697/) (2019)
- [Multimodal Model-Agnostic Meta-Learning via Task-Aware Modulation](https://mlsanity.com/paper/1910.13616/) (2019)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Task-adaptive Neural Process for User Cold-Start Recommendation](https://mlsanity.com/paper/2103.06137/) (2021)
- [Gradients as Features for Deep Representation Learning](https://mlsanity.com/paper/2004.05529/) (2020)
- [On Negative Interference in Multilingual Models: Findings and A Meta-Learning Treatment](https://mlsanity.com/paper/2010.03017/) (2020)
- [PID: Physics-Informed Diffusion Model for Infrared Image Generation](https://mlsanity.com/paper/2407.09299/) (2024)
- [A Survey on Machine Learning from Few Samples](https://mlsanity.com/paper/2009.02653/) (2020)
- [Hacking Task Confounder in Meta-Learning](https://mlsanity.com/paper/2312.05771/) (2023)
- [Neural Knitworks: Patched Neural Implicit Representation Networks](https://mlsanity.com/paper/2109.14406/) (2021)
- [PAC-Net: A Model Pruning Approach to Inductive Transfer Learning](https://mlsanity.com/paper/2206.05703/) (2022)
- [Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs](https://mlsanity.com/paper/2310.13270/) (2023)
- [NODE-ImgNet: a PDE-informed effective and robust model for image denoising](https://mlsanity.com/paper/2305.11049/) (2023)
- [Learning to Learn with Contrastive Meta-Objective](https://mlsanity.com/paper/2410.05975/) (2024)
- [Compositional meta-learning through probabilistic task inference](https://mlsanity.com/paper/2510.01858/) (2025)

## Built on by (most cited first, 1 of 1)

- [Solution-space heterogeneity shapes federated learning dynamics across partial differential equations](https://mlsanity.com/paper/2609.05012/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by embedding physical laws into the loss function. In parameterized PDE families, variations in coefficients or boundary/initial conditions define distinct tasks. This makes training individual PINNs for each task computationally prohibitive, while cross-task transfer can be sensitive to task heterogeneity. While meta-learning can reduce retraining cost, existing methods often rely on a single global initialization and may suffer from negative transfer, particularly under feature-scarce coordinate inputs and limited training-task availability. We propose the Learning-Affinity Adaptive Modular Physics-Informed Neural Network (LAM-PINN), a compositional framework that leverages task-specific learning dynamics. LAM-PINN combines PDE parameters with learning-affinity metrics from brief transfer sessions to construct a task representation and cluster tasks even with coordinate-only inputs. It decomposes the model into cluster-specialized subnetworks and a shared meta network, and learns routing weights to selectively reuse modules instead of relying on a single global initialization. Across three PDE benchmarks, LAM-PINN achieves an average 19.7-fold reduction in mean squared error (MSE) on unseen tasks using only 10% of the training iterations required by conventional PINNs. These results indicate its effectiveness for generalization to unseen configurations within bounded design spaces of parameterized PDE families in resource-constrained engineering settings.