Abstract

PDE surrogate models such as FNO and PINN struggle to predict solutions across inputs with diverse physical units and scales, limiting their out-of-distribution (OOD) generalization. We propose a π-invariant test-time projection that aligns test inputs with the training distribution by solving a log-space least squares problem that preserves Buckingham π-invariants. For PDEs with multidimensional spatial fields, we use geometric representative π-values to compute distances and project inputs, overcoming degeneracy and singular points that limit prior π methods. To accelerate projection, we cluster the training set into K clusters, reducing the complexity from O(MN) to O(KN) for the M training and N test samples. Across wide input scale ranges, tests on 2D thermal conduction and linear elasticity achieve MAE reduction of up to ≈91% with minimal overhead. This training-free, model-agnostic method is expected to apply to more diverse PDE-based simulations. 

A research team affiliated with UNIST has introduced a novel AI-based algorithm that enhances the accuracy of thermal and mechanical predictions across various scales, from microchips to large pipelines. 

Led by Professor Changwook Jeong from the Graduate School of Semiconductor Materials and Devices Engineering, their π-invariant test-time projection method realigns input data to conform with physical laws, addressing a crucial challenge in AI modeling—accurate predictions when faced with unfamiliar or out-of-distribution data.

The algorithm identifies the most physically similar data within existing training sets based on a dimensionless ratio derived from Buckingham's π theorem. It then transforms new inputs into familiar, physically consistent forms without retraining the model, operating in log space to preserve physical ratios. This approach is computationally efficient, reducing processing costs by approximately 99% compared to traditional methods.

Applied to 2D thermal conduction and linear elasticity problems, the technique achieved up to a 91% reduction in prediction error, even under conditions outside the original training range. It also demonstrated promising results in fluid dynamics, improving the accuracy of Navier–Stokes equation predictions in complex scenarios.

This advancement is expected to accelerate and economize simulations in semiconductor design, packaging reliability, battery management, and structural safety analysis—fields where varying sizes and conditions demand both precision and efficiency.

The study has been supported by the National Research Foundation of Korea (NRF) and the Institute of Information & Communications Technology Planning & Evaluation (IITP).

Journal Reference

Seokki Lee, Min-Chul Park, Giyong Hong, and Changwook Jeong, "Buckingham π-Invariant Test‑Time Projection for Robust PDE Surrogate Modeling," ICLR 2026 .