Diversity-Aware Adaptive Collocation for Physics-Informed Neural Networks via Sparse QUBO Optimization and Hybrid Coresets

2026-03-06

AI Summary

This work addresses the inefficiency of conventional physics-informed neural networks (PINNs), whose collocation strategies often oversample smooth regions, leading to redundant point sets and suboptimal training efficiency. The authors formulate collocation point selection as a subset selection problem that balances informativeness and diversity. They propose a sparse quadratic unconstrained binary optimization (QUBO) model based on a k-nearest neighbor similarity graph, integrating a linear term reflecting residual importance and a quadratic term penalizing redundancy. An exact budget-repair mechanism and a hybrid coreset strategy are introduced to maintain global PDE constraint coverage while strictly controlling the number of collocation points. Experiments on the one-dimensional Burgers’ equation with shock demonstrate that the method achieves comparable or higher accuracy under a fixed collocation budget while significantly reducing the computational cost of point selection.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-Informed Neural Networks (PINNs) enforce governing equations by penalizing PDE residuals at interior collocation points, but standard collocation strategies - uniform sampling and residual-based adaptive refinement - can oversample smooth regions, produce highly correlated point sets, and incur unnecessary training cost. We reinterpret collocation selection as a coreset construction problem: from a large candidate pool, select a fixed-size subset that is simultaneously informative (high expected impact on reducing PDE error) and diverse (low redundancy under a space-time similarity notion). We formulate this as a QUBO/BQM objective with linear terms encoding residual-based importance and quadratic terms discouraging redundant selections. To avoid the scalability issues of dense k-hot QUBOs, we propose a sparse graph-based BQM built on a kNN similarity graph and an efficient repair procedure that enforces an exact collocation budget. We further introduce hybrid coverage anchors to guarantee global PDE enforcement. We evaluate the method on the 1D time-dependent viscous Burgers equation with shock formation and report both accuracy and end-to-end time-to-accuracy, including a timing breakdown of selection overhead. Results demonstrate that sparse and hybrid formulations reduce selection overhead relative to dense QUBOs while matching or improving accuracy at fixed collocation budgets.

Problem

• Physics-Informed Neural Networks
• collocation points
• diversity-aware selection
• PDE residuals
• coreset construction

Innovation

• Physics-Informed Neural Networks
• Sparse QUBO Optimization
• Diversity-Aware Collocation
• Hybrid Coresets
• Adaptive Sampling