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

arXiv: https://arxiv.org/abs/2603.06761 · HTML: https://mlsanity.com/paper/2603.06761/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 6 Mar 2026.

## Builds on (most cited first, 13 of 13)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [GRAD-MATCH: Gradient Matching based Data Subset Selection for Efficient Deep Model Training](https://mlsanity.com/paper/2103.00123/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Introduction to Core-sets: an Updated Survey](https://mlsanity.com/paper/2011.09384/) (2020)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)

## Abstract (arXiv)

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.