PolyFormer: learning efficient reformulations for scalable optimization under complex physical constraints

TL;DR

PolyFormer introduces a physics-informed machine learning approach that reformulates complex constrained optimization problems into simpler polytopic forms, enabling faster and more scalable solutions without significant loss of accuracy.

Contribution

It presents PolyFormer, a novel method that leverages geometric structures to transform complex physical constraints into efficient reformulations for scalable optimization.

Findings

Achieves up to 6,400-fold speedup in computation

Reduces memory usage by up to 99.87%

Maintains or improves solution quality compared to state-of-the-art methods

Abstract

Real-world optimization problems are often constrained by complex physical laws that limit computational scalability. These constraints are inherently tied to complex regions, and thus learning models that incorporate physical and geometric knowledge, i.e., physics-informed machine learning (PIML), offer a promising pathway for efficient solution. Here, we introduce PolyFormer, which opens a new direction for PIML in prescriptive optimization tasks, where physical and geometric knowledge is not merely used to regularize learning models, but to simplify the problems themselves. PolyFormer captures geometric structures behind constraints and transforms them into efficient polytopic reformulations, thereby decoupling problem complexity from solution difficulty and enabling off-the-shelf optimization solvers to efficiently produce feasible solutions with acceptable optimality loss. Through evaluations across three important problems (large-scale resource aggregation, network-constrained optimization, and optimization under uncertainty), PolyFormer achieves computational speedups up to 6,400-fold and memory reductions up to 99.87%, while maintaining solution quality competitive with or superior to state-of-the-art methods. These results demonstrate that PolyFormer provides an efficient and reliable solution for scalable constrained optimization, expanding the scope of PIML to prescriptive tasks in scientific discovery and engineering applications.