Minsu Koh

Beomchul Park, Minsu Koh, Heejo Kong et al.

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.

Minsu Koh, Beom-Chul Park, Heejo Kong et al.

Neural operators have emerged as promising frameworks for learning mappings governed by partial differential equations (PDEs), serving as data-driven alternatives to traditional numerical methods. While methods such as the Fourier neural operator (FNO) have demonstrated notable performance, their reliance on uniform grids restricts their applicability to complex geometries and irregular meshes. Recently, Transformer-based neural operators with linear attention mechanisms have shown potential in overcoming these limitations for large-scale PDE simulations. However, these approaches predominantly emphasize global feature aggregation, often overlooking fine-scale dynamics and localized PDE behaviors essential for accurate solutions. To address these challenges, we propose the Locality-Aware Attention Transformer (LA2Former), which leverages K-nearest neighbors for dynamic patchifying and integrates global-local attention for enhanced PDE modeling. By combining linear attention for efficient global context encoding with pairwise attention for capturing intricate local interactions, LA2Former achieves an optimal balance between computational efficiency and predictive accuracy. Extensive evaluations across six benchmark datasets demonstrate that LA2Former improves predictive accuracy by over 50% relative to existing linear attention methods, while also outperforming full pairwise attention under optimal conditions. This work underscores the critical importance of localized feature learning in advancing Transformer-based neural operators for solving PDEs on complex and irregular domains.