Neural PDE solver improves accuracy with joint spectral physical modeling
Transolver-$σ$: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving
Artificial Intelligence
Summary
Simulating complex physical systems often involves solving partial differential equations (PDEs), which can be slow and tricky over time. The authors found that models focusing only on physical states or only on spectral (frequency) information each have strengths and weaknesses in accuracy over time. They created Transolver-σ, a new neural network that combines both physical and spectral views inside its workings, improving long-term prediction accuracy. This method reduced errors significantly across multiple PDE problems and worked well on complex real-world fluid and combustion data.
What this means in practice
- •For computational engineers: Enhance simulation accuracy and efficiency for fluid and combustion PDEs by integrating physical and spectral data representations.
- •For industrial simulation teams: Improve predictive reliability for coupled multiphysics systems in engineering workflows using the joint spectral-physical neural PDE solver.
Authors
Haonan Shangguan, Hang Zhou, Haixu Wu, Yuezhou Ma, Jianmin Wang, Mingsheng Long
Abstract
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.