Reliable high-resolution state estimates for forecasts and reanalyses are pivotal in environmental applications, particularly in ocean and atmospheric sciences. These are typically achieved by integrating observational data into dynamical models through processes such as data assimilation (DA), when enhancing the reliability of forecasts and reanalysis, or downscaling when bridging the gap between coarse-scale observations and fine-scale information. Current DA and downscaling techniques rely on limiting assumptions and tend to be computationally demanding, especially in the presence of observational and model uncertainties. Artificial intelligence (AI) emerges as a powerful avenue for developing efficient data-driven tools that enhance reliability and alleviate computational demands of conventional DA and downscaling algorithms. This talk aims to present recent developments in AI tools that address challenges pertaining to downscaling with application to chaotic dynamical systems, and within an uncertain framework. The state-of-the-art dynamical downscaling algorithm, Continuous data assimilation (CDA), and its discrete-in-time counterpart (DDA) are first explored in the setting involving observational errors. Since CDA relies on an abstract lifting function called the determining form map, a physics-informed deep neural network (PI-DNN) named CDAnet is proposed to approximate this intractable mapping. CDAnet is then evaluated under observational and model uncertainties in application to the Rayleigh-Benard convection problem, validating and further extending upon the knowledge from theory.

Several recent studies considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result about relations between non-local and local Cahn-Hilliard, we also derive rigorously the large-friction nonlocal- to-local limit. The result is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. This approach provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation. During the talk I will also discuss the high-friction limit of the Euler-Poisson system.

Fractional diffusion equations (FDEs) are powerful tools for modeling anomalous diffusion in complex systems, such as fractured media and biological processes, where nonlocal dynamics and spatial heterogeneity are prominent. These equations provide a more accurate representation of such systems compared to classical models but pose significant computational challenges, particularly for spatially varying diffusivity and fractional orders. In this talk I will present a Bayesian inverse problem for FDEs in a 2-dimensional bounded domain with an anomaly of unknown geometric and physical properties, where the latter are the diffusivity and fractional order fields. To tackle the computational burden of solving dense and ill-conditioned systems, we employ an advanced finite-element scheme incorporating low-rank matrix representations and hierarchical matrices. For parameter estimation, we implement two surrogate-based approaches using polynomial chaos expansions: one constructs a 7-dimensional surrogate for simultaneous inference of geometrical and physical parameters, while the other leverages solution singularities to separately infer geometric features, then constructing a 2-dimensional surrogate to learn the physical parameters and hence reducing the computational cost immensely.

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