We made substantial progress across all three scientific objectives of SpaTe. In particular, we developed new space–time representations and scalable surrogate models for complex physical dynamics, including transformer- and diffusion-based architectures and robust similarity metrics that characterize the structure of 4D simulation data.
In parallel, we advanced the project’s core vision of differentiable physics for inverse problems, releasing high-performance differentiable solvers (including a GPU-accelerated multi-block PISO solver), establishing stable long-horizon optimization techniques, and demonstrating unsupervised 4D reconstruction of flow fields from sparse observations. For unstructured and Lagrangian data, we introduced physically constrained particle-learning methods and symmetry-aware operators that form the basis for adaptive Lagrangian representations.
Across all lines, we created and disseminated open-source software, datasets, and benchmarking tools (e.g. differentiable turbulence models, APEBench, PDE-Transformer/P3D code bases), enabling broad adoption by the fluid-mechanics and machine-learning communities. Together, these contributions not only fulfil the scientific objectives of SpaTe but also establish strong foundations for next-generation simulation tools, data-driven design workflows, and hybrid AI-physics modelling approaches.
During the project, we disseminated our results widely across leading journals and conferences in machine learning, computational physics, and fluid mechanics. Our advances in space–time representations, stability analysis, and neural surrogate modelling were published at ICLR, ICML, NeurIPS, CMAME, NCAA, and related ML workshops. Our core contributions on differentiable solvers, turbulence modelling, and inverse problems appeared in top archival venues such as Journal of Fluid Mechanics, Journal of Computational Physics, Physical Review E, Computer Methods in Applied Mechanics and Engineering, as well as NeurIPS and ICLR for methodological breakthroughs. Our work on Lagrangian and unstructured data learning was disseminated through ICLR, NeurIPS (including an oral), and CVPR. Additionally, we released open benchmarks and tools (e.g. APEBench, PDE-Transformer, differentiable turbulence solvers) through NeurIPS, ICLR, ICML, and public code repositories, ensuring strong visibility, community uptake, and long-term impact.
