SISSA Trieste - Italy
Conservation laws lie at the heart of CFD and many multiphysics applications. Yet many real-world workflows, such as design optimisation, control, uncertainty-aware decision making and emerging digital-twin pipelines, remain constrained by the cost of resolving multiscale physics at high fidelity. Building reduced-order models (ROMs) in this setting is notoriously challenging: nonlinear fluxes, sharp gradients and shocks, evolving geometries, and parameter variations can quickly destroy stability and compromise key structural properties, making their preservation far from trivial even for well-established numerical schemes. We outline a hybrid-surrogate perspective that systematically fuses classical Reduced Order Modelling (ROM) with Scientific Machine Learning (SciML) to deliver fast, robust, and physically consistent reduced models. We will first present a non-intrusive strategy based on space-dependent aggregation of reduced models: instead of committing to a single reduction/regression pipeline, multiple ROM surrogates are constructed and locally blended using spatially varying convex weights learned from data. This mixture-of-surrogates adapts across regimes within the same configuration and improves predictive performance on demanding CFD benchmarks. Next, we introduce a complementary non-intrusive approach based on machine-learning quadratic closures for under-resolved POD-ROMs. The idea is to re-inject the contribution of truncated modes through a quadratic correction term acting on the POD coefficients yielding a discretisation-agnostic, parameter-dependent closure with improved generalisation I will then move to intrusive reduced models and the central issue of closure across parameter variations. Here, operator-learning components, particularly deep operator networks, learn reduced correction terms that represent the impact of unresolved scales and inject them into a POD–Galerkin backbone, improving accuracy and stabilising predictive regimes while retaining the efficiency and interpretability of projection-based ROMs. Finally, we do frame these advances within a roadmap for trustworthy surrogates for digital twins built around four main goals: handling physics and geometry complexity; learning effectively from scarce/noisy/partial observations; enforcing structure as a first-class design constraint; and supporting rigorous verification and validation.