Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes

We propose Hermite-NGP, a gradient-augmented multi-resolution hash encoding designed to enable fast and accurate computation of spatial derivatives for neural PDE solvers. Unlike existing NGP-based approaches that rely on automatic differentiation or finite differences and suffer from instability or high cost, Hermite-NGP explicitly stores function values and mixed partial derivatives at hash grid vertices, allowing fully analytic evaluation of gradients, Jacobians, and Hessians via Hermite interpolation. This design preserves the efficiency and spatial adaptivity of NGP while supporting analytic differential operators up to second order. We further introduce a multi-resolution curriculum training strategy analogous to multigrid V-cycles to enable coarse-to-fine optimization. Across a range of 2D and 3D PDE benchmarks, Hermite-NGP achieves up to approximately 20 times lower error than prior neural PDE methods, and reduces wall-clock convergence time by 2 to 10 times compared to other solvers, with per-epoch training times as low as 3.5 ms for models with up to 17M parameters.

Scientific machine learning methods such as physics-informed neural networks (PINNs) increasingly rely on domain decomposition for better scalability while solving partial differential equations (PDEs) over complex geometries, yet the resulting composite loss comprising residual, boundary, and interface terms is highly susceptible to conflicting gradients that degrade training. This work bridges domain decomposition with projection-based gradient surgery to systematically mitigate such conflicts in 2D and 3D settings. We evaluate two existing projection-based algorithms, PCGrad and ConFIG, and identify their performance degradation in specific scenarios such as 3D domains with multiple overlapping interfaces. To address this limitation, we propose Norm-PCGrad, a normalized variant that achieves state-of-the-art accuracy across a range of 2D and 3D domain decomposition problems. Across the benchmarks considered, Norm-PCGrad consistently achieves the lowest relative $L_2$ error compared to training without gradient surgery as well as to existing algorithms such as PCGrad and ConFIG, while incurring negligible additional computational overhead. To improve computational efficiency of domain decomposition frameworks such as Extended PINN (XPINN), we propose replacing vanilla PINNs in selected subdomains with separable architectures such as Separable PINN (SPINN), reducing the computational cost from quadratic (or cubic) to linear. We additionally demonstrate that gradient surgery extends to physics-informed Kolmogorov-Arnold Networks (PIKANs), yielding substantial accuracy improvements for 3D domain decomposition and confirming the generality of the proposed approach across network architectures.

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.