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

• URL: http://arxiv.org/abs/2605.15254v1
• Date: Thu, 14 May 2026 17:16:04 GMT
• Title: Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation
• Abstract summary: We propose a correlated curriculum learning framework for spatially separated subregions.<n>This is the first work to address difficulties from the perspective of subregions.
• Score: 5.585973913525636
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Multi-dimensional training-priority weighting based on physical information propagation paths: a unified residual-weighting framework for physics-informed neural networks [7.031909944831412]
  Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs)<n>PINNs' synchronous optimization treats residuals of different regions and constraints equally, which is inconsistent with the progressive "from source to response" physical information propagation path, degrading training stability and accuracy.<n>We propose a unified multi-dimensional priority-constraint framework that partitions the domain along the propagation path and constructs negative-exponential residual weights, converting the physical propagation order into a training priority.
  arXiv  Detail & Related papers  (2026-07-13T05:02:07Z)
• Dynamics and Representation Structure of Local Approximations to Gradient-Based Learning in Linear Recurrent Neural Networks [56.42974363744522]
  We study the learning dynamics of random feedback local online (RFLO) and truncated backpropagation through time (tBPTT)<n>Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.
  arXiv  Detail & Related papers  (2026-05-29T18:19:45Z)
• When Learning Hurts: Fixed-Pole RNN for Real-Time Online Training [58.25341036646294]
  We analytically examine why learning recurrent poles does not provide tangible benefits in data and empirically offer real-time learning scenarios.<n>We show that fixed-pole networks achieve superior performance with lower training complexity, making them more suitable for online real-time tasks.
  arXiv  Detail & Related papers  (2026-02-25T00:15:13Z)
• Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method [1.2289361708127877]
  Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs)<n>Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs.<n>We propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions.
  arXiv  Detail & Related papers  (2025-01-13T21:24:15Z)
• Online Pseudo-Zeroth-Order Training of Neuromorphic Spiking Neural Networks [69.2642802272367]
  Brain-inspired neuromorphic computing with spiking neural networks (SNNs) is a promising energy-efficient computational approach. Most recent methods leverage spatial and temporal backpropagation (BP), not adhering to neuromorphic properties. We propose a novel method, online pseudo-zeroth-order (OPZO) training.
  arXiv  Detail & Related papers  (2024-07-17T12:09:00Z)
• DeltaPhi: Physical States Residual Learning for Neural Operators in Data-Limited PDE Solving [54.605760146540234]
  DeltaPhi is a novel learning framework that transforms the PDE solving task from learning direct input-output mappings to learning the residuals between similar physical states.<n>Extensive experiments demonstrate consistent and significant improvements across diverse physical systems.
  arXiv  Detail & Related papers  (2024-06-14T07:45:07Z)
• Towards Interpretable Deep Local Learning with Successive Gradient Reconciliation [70.43845294145714]
  Relieving the reliance of neural network training on a global back-propagation (BP) has emerged as a notable research topic. We propose a local training strategy that successively regularizes the gradient reconciliation between neighboring modules. Our method can be integrated into both local-BP and BP-free settings.
  arXiv  Detail & Related papers  (2024-06-07T19:10:31Z)
• Neural Network with Local Converging Input (NNLCI) for Supersonic Flow Problems with Unstructured Grids [0.9152133607343995]
  We develop a neural network with local converging input (NNLCI) for high-fidelity prediction using unstructured data. As a validation case, the NNLCI method is applied to study inviscid supersonic flows in channels with bumps.
  arXiv  Detail & Related papers  (2023-10-23T19:03:37Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions [22.514769448363754]
  Training challenges in PINNs and XPINNs for time-dependent PDEs are discussed. We propose a new stacked-decomposition method that bridges the gap between PINNs and XPINNs. We also formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality.
  arXiv  Detail & Related papers  (2023-02-28T01:19:21Z)
• Less Emphasis on Difficult Layer Regions: Curriculum Learning for Singularly Perturbed Convection-Diffusion-Reaction Problems [21.615494601195472]
  We show that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We propose a novel curriculum learning method that encourages neural networks to prioritize learning on easier non-layer regions while downplaying learning on harder layer regions.
  arXiv  Detail & Related papers  (2022-10-23T10:15:32Z)
• Adaptive Self-supervision Algorithms for Physics-informed Neural Networks [59.822151945132525]
  Physics-informed neural networks (PINNs) incorporate physical knowledge from the problem domain as a soft constraint on the loss function. We study the impact of the location of the collocation points on the trainability of these models. We propose a novel adaptive collocation scheme which progressively allocates more collocation points to areas where the model is making higher errors.
  arXiv  Detail & Related papers  (2022-07-08T18:17:06Z)
• Optimal Transport Based Refinement of Physics-Informed Neural Networks [0.0]
  We propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT) PINNs solvers have been found to suffer from a host of issues: spectral bias in fully-connected pathologies, unstable gradient, and difficulties with convergence and accuracy. We present a novel training strategy for solving the Fokker-Planck-Kolmogorov Equation (FPKE) using OT-based sampling to supplement the existing PINNs framework.
  arXiv  Detail & Related papers  (2021-05-26T02:51:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.