# Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation

arXiv: https://arxiv.org/abs/2605.15254 · HTML: https://mlsanity.com/paper/2605.15254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 14 May 2026.

## Builds on (most cited first, 10 of 10)

- [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)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Training](https://mlsanity.com/paper/2211.11396/) (2022)
- [Frequency-adaptive Multi-scale Deep Neural Networks](https://mlsanity.com/paper/2410.00053/) (2024)
- [Long-term simulation of physical and mechanical behaviors using curriculum-transfer-learning based physics-informed neural networks](https://mlsanity.com/paper/2502.07325/) (2025)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) combine deep learning with physical constraints for solving partial differential equations (PDEs), and are widely applied in fluid mechanics, heat transfer, and solid mechanics. However, PINN training still suffers from high-dimensional non-convex loss landscapes, imbalanced multiobjective constraints, and ineffective information propagation. Existing curriculum learning and causality-guided strategies improve training stability, but mainly focus on temporal or parametric progression, lacking explicit treatment of spatial information propagation and inter-region consistency. Moreover, they are not directly applicable to boundary value problems (BVPs) with strong spatial coupling. To address this issue, we propose a spatially correlated curriculum learning framework for PINNs. To the best of our knowledge, this is the first work to address PINN training difficulties from the perspective of spatial coupling among subregions. First, spatial causal weights guide information from near-boundary regions inward, reducing optimization failures and spurious convergence. Second, a low-frequency information bridge enforces pseudo-label-based consistency across spatially separated regions, suppressing global low-frequency drift. Third, a region-adaptive reweighting strategy adjusts subregion losses to reduce local residuals and recover high-frequency details. Experiments on PDE benchmarks show that, under comparable computational cost, the proposed method alleviates training failures and improves solution accuracy. The code is available at https://github.com/pigofmomo/CurriculumLearningPINN.