Transolver-$σ$: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

2026-09-29

AI Summary

This study addresses the error accumulation problem in autoregressive rollout of neural PDE solvers, despite their high single-step accuracy, by proposing a solution framework based on joint spectral and physical subspace modeling. Methodologically, a dedicated latent space is designed to enable complementary dual-modality information fusion. A sliced residual physics attention mechanism is introduced to preserve identity paths, combined with an axis-factorized Fourier operator to capture global structures. Furthermore, adaptive physics-state interaction and joint subspace reconstruction are employed to enhance solving stability. Experiments demonstrate that the proposed method reduces relative errors by 33.4% across five benchmarks, significantly improving long-term autoregressive prediction accuracy for multiphysics coupled systems and real-world fluid measurements.

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.

Problem

• Neural PDE Solvers
• Autoregressive Rollout
• Spectral-Physical Subspace
• Time-dependent PDEs

Innovation

• Neural PDE Solver
• Joint Spectral-Physical Subspace Modeling
• Slice-Residual Physics-Attention
• Axis-Factorized Fourier Operator
• Autoregressive Rollout