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 $a_t$ denote the direction constructed by gradient surgery, $u_t$ the optimizer proposal, and $\mathcal{C}_t$ the conflict-free cone induced by the loss-specific gradients. We show that modern optimizers can transform $a_t$ through mechanisms such as historical state, adaptive scaling, preconditioning, or decoupled weight decay, so $a_t \in \mathcal{C}_t$ does not generally imply $u_t \in \mathcal{C}_t$. We refer to this optimizer-induced discrepancy in conflict-freeness between $a_t$ and $u_t$ as Gradient-Update Mismatch (GUM). Accordingly, we propose Gradient-Update Alignment (GUA), which projects $u_t$ onto $\mathcal{C}_t$ to obtain the aligned update $p_t$ and applies $p_t$ 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 $L_2$ error by up to 98.2% in individual settings. Data and code are available at https://github.com/JingXiao10/GUA.
Jing Xiao, Xinhai Chen, Qinglin Wang et al.· 0 citations
Data-driven learning provides a promising route for accelerating computational fluid dynamics, yet transient flow prediction remains challenging because models must simultaneously capture evolving temporal dynamics and cross-scale spatial structures while generalizing across geometries and operating conditions. In this work, we study one-step-ahead prediction of two-dimensional incompressible transient flows from historical snapshots and propose Multi-Scale Spatiotemporal Flow (MuST-Flow), a physics-informed multi-scale spatiotemporal network for recurrent forecasting of velocity and pressure fields. The central idea of MuST-Flow is to couple cross-scale feature extraction with physical regularization, enabling the model to better resolve localized high-gradient flow structures while maintaining physically consistent evolution. Specifically, MuST-Flow adopts a multi-scale spatiotemporal convolutional design with dilated receptive fields to learn hierarchical flow representations and incorporates a hybrid loss derived from the residuals of the incompressible Navier–Stokes equations to encourage mass and momentum consistency during training. To evaluate generalization beyond fixed configurations, the proposed method is assessed on transient hydrofoil flow data under cross-geometry testing with varied operating conditions. Experimental results show that MuST-Flow achieves the lowest mean absolute error, highest structural similarity, and lowest speed and velocity-direction errors among both generic spatiotemporal prediction baselines and flow-oriented neural-operator baselines, while maintaining competitive divergence and Navier–Stokes residuals. These results demonstrate its effectiveness as an efficient surrogate modeling framework for two-dimensional incompressible transient flow prediction.
Yan Liu, Jie Liu, Xinhai Chen et al.· The Physics of Fluids· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
