Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.
Transformer architectures have attracted increasing attention for solving partial differential equations (PDEs), owing to their flexibility in handling irregular discretizations and their ability to capture long-range physical dependencies. However, unlike discrete language tokens or fixed-resolution image patches, observed physical fields are finite samples of underlying infinite-dimensional functions. Consequently, effectively applying Transformers to PDEs requires a tokenizer that respects the functional nature of physical fields and constructs physically expressive tokens from arbitrary discretizations.To this end, we propose \methodname{Physics Transformer}, a function-projection-based Transformer architecture for physical field prediction. Physics Transformer treats a physical field as a continuous function and partitions its discretization into locality-preserving spatial patches. Within each patch, it dynamically learns a set of adaptive local basis functions and projects the sampled field onto these bases to obtain compact physics tokens. The resulting tokens capture diverse latent physical states while preserving fine-scale spatial structures, enabling efficient global interaction through factorized attention across space and physical states. The projected representation further supports efficient decoding at arbitrary query locations. Extensive experiments on diverse benchmarks, ranging from two-dimensional PDE dynamics to industrial-scale three-dimensional CFD simulations, demonstrate that Physics Transformer accurately captures fine-grained physical structures and achieves state-of-the-art predictive performance. These results establish function projection as a practical and effective foundation for designing Transformer architectures for PDE solving.
Predicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address these challenges is leveraging physics priors in training neural networks, known as physics-informed deep learning (PiDL). In this work, we introduce the Multi-Resolution Finite-Volume-inspired network, MuRFiV, designed to capitalize on the conservative property of finite volume on the global scale and the expressive power of deep learning on the local scale. We demonstrate the effectiveness of MuRFiV on several spatio-temporal systems governed by partial differential equations (PDEs), including Burgers' equation, shallow water equations, and incompressible Navier-Stokes equations. By embedding PDE information into the deep learning architecture, MuRFiV achieves strong long-term prediction accuracy and remains stable over very long autoregressive rollouts, significantly outperforming data-driven neural network baselines. This result highlights the promise of combining multiresolution learning with finite-volume-inspired inductive bias for accurate and robust long-term prediction of complex dynamics.