发表机构
Tsinghua University; MIT CSAIL（清华大学; 麻省理工学院计算机科学与人工智能实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
Transolver-$\sigma$提出联合谱-物理子空间建模的神经PDE求解器，通过SRPA和傅里叶算子协同，在五个基准上平均相对误差降低33.4%，并改善自回归展开。
AI 中文摘要
神经求解器为偏微分方程（PDE）的数值模拟提供了高效的替代方案。对于时间相关问题，强的一步精度并不一定能转化为可靠的自回归展开。我们观察到，仅基于物理状态建模的求解器可以实现更低的一步误差，而其仅基于谱的对应求解器在后续展开步骤中可能变得更加准确。受此观察启发，我们提出了Transolver-$\sigma$，一种基于联合谱-物理子空间建模的神经PDE求解器。在每个块内，自适应物理状态交互和谱变换在专用的潜在子空间中进行建模，其响应被重新组合以实现两种表示之间的信息交换。在物理子空间内，我们引入了切片残差物理注意力（SRPA），它保留了显式的切片空间身份路径，同时保持可学习的跨切片交互。并行地，一个轴因子化的傅里叶算子捕获全局谱结构。在跨越稳态预测和时间依赖动力学的五个公认PDE基准上，Transolver-$\sigma$实现了最先进的性能，与每个指标的最强基线相比，基准平均相对误差降低了33.4%，同时一致地改善了自回归展开，优于单算子对应方法。Transolver-$\sigma$还在耦合多物理系统以及来自RealPDEBench的真实世界流体和燃烧测量上带来了显著的提升，证明了其在标准模拟基准之外的有效性。
英文摘要
Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can achieve lower one-step error, whereas its spectral-only counterpart can become more accurate at later rollout steps. Motivated by this observation, we present Transolver-$σ$, a neural PDE solver based on joint spectral--physical subspace modeling. Within each block, adaptive physical-state interactions and spectral transformations are modeled in dedicated latent subspaces, whose responses are recomposed to enable information exchange between the two representations. Within the physical subspace, we introduce Slice-Residual Physics-Attention (SRPA), which preserves an explicit slice-space identity path while retaining learnable cross-slice interaction. In parallel, an axis-factorized Fourier operator captures global spectral structure. Across five well-established PDE benchmarks spanning steady-state prediction and time-dependent dynamics, Transolver-$σ$ achieves state-of-the-art with a benchmark-averaged relative error reduction of 33.4% over the strongest baseline for each metric, while consistently improving autoregressive rollout over single-operator counterparts. Transolver-$σ$ further delivers strong gains on coupled multiphysics systems and real-world fluid and combustion measurements from RealPDEBench, demonstrating its effectiveness beyond standard simulation benchmarks.
