Gradient-Update Mismatch: Rethinking Conflict-Free Training of Physics-Informed Neural Networks

Training Physics-Informed Neural Networks (PINNs) requires jointly optimizing physics residual and initial/boundary condition loss terms, which often induce conflicting gradients. Gradient surgery methods mitigate this issue by constructing directions from loss-specific gradients to reduce conflict before optimizer transformation. However, even when the constructed direction is conflict-free, this property may not be preserved after optimizer transformation. Let at denote the direction constructed by gradient surgery, ut the optimizer proposal, and Ct the conflict-free cone induced by the loss-specific gradients. We show that modern optimizers can transform at through mechanisms such as historical state, adaptive scaling, preconditioning, or decoupled weight decay, so at ∈ Ct does not generally imply ut ∈ Ct. We refer to this optimizer-induced discrepancy in conflict-freeness between at and ut as Gradient-Update Mismatch (GUM). Accordingly, we propose Gradient-Update Alignment (GUA), which projects ut onto Ct to obtain the aligned update pt and applies pt to the parameters. When the optimizer maintains internal state, GUA further adjusts this state toward targets reconstructed from the applied update. We conduct extensive experiments and find that GUM is widespread across momentum, adaptive, and curvature-based optimizers, with conflict rates reaching up to 86.3%. Across all PINN settings, GUA achieves conflict-free applied updates and consistently improves various gradient surgery methods, reducing the relative L2 error by up to 98.2% in individual settings. Data and code are available at https://github.com/JingXiao10/GUA.