PINN loss functions: why physics-informed networks often fail to train

2026-05-20 · Source: Machine Learning ML & Generative AI News · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Advanced, quick

Summary

Physics-Informed Neural Networks (PINNs) utilize known Partial Differential Equation (PDE) constraints alongside data to approximate unknown functions, aiming for faster convergence and better generalization than data-only models. However, their training is notoriously difficult due to complex loss functions, which are weighted sums of multiple terms like PDE residuals, boundary conditions, initial conditions, and data. Research by Wang, Teng & Perdikaris (2021) shows severe gradient imbalance among loss components, causing optimizers to prioritize louder gradients. Wang, Yu & Perdikaris (2022) used Neural Tangent Kernel theory to demonstrate that PDE residual terms have much smaller eigenvalues, leading to rapid learning of boundaries but slow learning of interior physics. Krishnapriyan et al. (NeurIPS 2021) further illustrated that PINNs systematically fail to converge on simple PDEs, such as the convection equation, as the convection coefficient increases, even with reasonable hyperparameters. While mitigations like adaptive loss weighting, causal training, curriculum approaches, and architectural fixes exist, none have fully resolved these training pathologies.

Key takeaway

For Machine Learning Engineers or Research Scientists developing Physics-Informed Neural Networks, you should anticipate significant training challenges stemming from loss function imbalances. If your PINN struggles to converge, investigate gradient magnitudes across loss components and consider implementing adaptive weighting schemes, causal training, or architectural modifications to hard-code boundary conditions. These strategies can help mitigate the observed pathologies where boundary conditions learn quickly but interior physics lags, preventing robust model convergence.

Key insights

PINNs struggle to train due to severe gradient imbalance and scale differences among their multi-term loss functions.

Principles

• Gradient imbalance causes optimizers to prioritize "loudest" loss components.
• PDE residual terms often have smaller eigenvalues than boundary losses.
• PINNs can fail to converge on simple PDEs with increasing coefficients.

In practice

• Explore adaptive loss weighting for PINN training.
• Consider causal training or curriculum learning approaches.
• Investigate architectural fixes for boundary conditions.

Topics

• Physics-Informed Neural Networks
• PINN Training
• Loss Functions
• Gradient Imbalance
• Partial Differential Equations
• Neural Tangent Kernel

Best for: AI Engineer, AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning ML & Generative AI News.