[
    {
        "id": "osp-20584",
        "type": "article-journal",
        "title": "Transolver-$σ$: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving",
        "author": [
            {
                "family": "Shangguan",
                "given": "Haonan"
            },
            {
                "family": "Zhou",
                "given": "Hang"
            },
            {
                "family": "Wu",
                "given": "Haixu"
            },
            {
                "family": "Ma",
                "given": "Yuezhou"
            },
            {
                "family": "Wang",
                "given": "Jianmin"
            },
            {
                "family": "Long",
                "given": "Mingsheng"
            }
        ],
        "URL": "https://omanscience.com/ar/articles/transolver-joint-spectral-physical-subspace-modeling-for-neural-pde-solving",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "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."
    }
]