Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors during long-horizon autoregressive prediction and may fail to exploit available governing-equation structure. Existing approaches incorporate physics primarily through residual-based training objectives or PDE-specific architectural constraints, which can introduce optimization difficulties or limit architectural generality. In this work, we introduce a representation-level approach to physics integration in which a feed-forward Gaussian splatting (FFGS) representation serves as a continuous interface between discretized solution fields and governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field with closed-form spatial derivatives, allowing available physical PDE operators to be integrated directly within the learned evolution map without introducing a physics-residual loss. We evaluate the framework across two- and three-dimensional PDE systems, including advection, diffusion, nonlinear self-advection, and reaction dynamics. Over long-horizon autoregressive rollouts, the proposed framework reduces relative $\ell_2$ error by $1.5\times$--$2.2\times$ compared with the strongest purely data-driven baseline across the benchmark suite, while consistently improving spectral fidelity. The framework also remains effective when the governing equations are partially known, demonstrating robustness to incomplete physics. These results demonstrate that continuous field representations can provide a practical interface for incorporating known physical structure into generic neural-operator surrogates.
Accurate, patient-specific vascular geometry is pivotal for diagnosis, planning, and device design, yet existing statistical shape modeling (SSM) pipelines rely on linear priors and topology-specific preprocessing that limit realism, scalability, and interoperability. We present HUG-VAS, a Hierarchical NURBS Generative framework for Vascular models, that unifies NURBS-based 3D shape encoding with diffusion-based generative modeling to synthesize fine-grained, CFD-ready aortic anatomies. HUG-VAS factorizes shape into (i) vessel centerlines generated by a denoising diffusion model and (ii) cross-sectional radius profiles synthesized by a classifier-free guided diffusion model conditioned on the centerline, thereby decoupling and preserving stochastic variability across these two anatomical layers. Beyond unconditional synthesis, we enable training-free, zero-shot conditional generation via diffusion posterior sampling from image-derived prompts (e.g., sparse 3D points, slice contours, or partial surface patches), supporting interactive semi-automatic segmentation, editing and robust reconstruction under degraded imaging. Trained on 21 patient-specific MRA cases, HUG-VAS generates multi-branch aortas with supra-aortic vessels whose biomarker distributions closely match the source cohort, and whose watertight NURBS outputs directly integrate with downstream CFD solvers. To our knowledge, HUG-VAS is the first SSM frameworks to unify NURBS parameterization, hierarchical diffusion, and DPS-based zero-shot conditional generation, enabling reconstruction and completion of vascular geometry from sparse, partial geometric priors without retraining.
Pan Du, Ming-Qi Xu, Xiaozhi Zhu et al.· Computer Methods in Applied...· 0 citations
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