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

2026-04-28

AI Summary

This work addresses the poor generalization and high training cost of conventional physics-informed neural networks (PINNs) when transferring across heterogeneous parametric partial differential equation (PDE) tasks. The authors propose LAM-PINN, a novel framework that introduces, for the first time, a learnable affinity-based task clustering mechanism to construct PDE parameter-driven task representations. LAM-PINN employs a modular architecture that decomposes the network into a shared meta-network and cluster-specific subnetworks, enabling module reuse through dynamic routing. Requiring only coordinate inputs and seamlessly integrating physical constraints with data-driven learning, the method achieves an average 19.7× reduction in mean squared error on unseen tasks across three PDE benchmarks, using merely 10% of the standard training iterations.

Abstract

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.

Problem

• task heterogeneity
• physics-informed neural networks
• meta-learning
• parameterized PDEs
• negative transfer

Innovation

• compositional meta-learning
• task heterogeneity
• physics-informed neural networks
• learning-affinity metrics
• modular architecture