Mitigating Gradient Competition and Capturing PDE Frequency in PINNs

Develop training methodologies and loss formulations for physics-informed neural networks (PINNs) that mitigate gradient competition among multiple objective terms in the loss function and enable accurate representation of the frequency content of the governing partial differential equations. The goal is to ensure stable and accurate training across coupled and multi-physics PDE systems where multiple losses (e.g., PDE residuals, boundary/interface conditions) interact, especially under extreme physical parameter regimes.

Background

The paper studies coupled Stokes–Darcy systems using physics-informed neural networks and shows that extreme physical constants can create ill-conditioned multi-objective losses, leading to gradient competition and training failures.

While the authors introduce MF-PINNs—combining velocity–pressure and stream–vorticity forms with weighted losses—to alleviate these issues and demonstrate improvements, they emphasize that the broader challenge of systematically resolving gradient competition while accurately capturing the frequency content of PDE solutions remains unresolved.

References

However, how to mitigate the gradient competition between multi-objective loss functions and accurately capture the frequency of PDEs remains an open research question.

— A Mixed-Form PINNS (MF-PINNS) For Solving The Coupled Stokes-Darcy Equations  (2510.17508 - Shan et al., 20 Oct 2025) in Section 1: Introduction

An open question is whether this complementarity generalizes beyond our controlled benchmark, to three-dimensional, higher-Reynolds-number flows where the underlying dynamics are far more complex, and where techniques validated here may compose differently, or not at all.

— Complementary, Not Cumulative: Interaction Effects in Physics-Informed Neural Networks for Navier-Stokes Vortex Shedding  (2608.19632 - Shah, 20 Aug 2026) in Section 6, Conclusion, p. 10

We regard removing the PINN's remaining dependence on hand-tuned weights, whether through a better-conditioned network architecture, a more informative choice of collocation points, or a principled rather than hand-tuned weighting of the consistency terms, as an important open problem rather than a solved part of the present framework.

— Nonperturbative functional renormalization group for Higgs-singlet models with physics-informed neural networks  (2609.08470 - Yokozaki, 8 Sep 2026) in Section 5.4, “Practical role and limitations of the soft consistency penalty” (also reiterated in Section 6)

We conjecture that in all cases the term above is one of the main driving directions.

— Operator-informed initialization for Fourier features physics-informed neural networks  (2610.03378 - Molina et al., 2 Oct 2026) in Remark immediately following equation (the residual dynamics equation), Section 2.2, “Neural network model and training dynamics”