Many modern data science applications rely on data from distributed, autonomous, and confidential sources, including environmental monitoring systems, urban sensing platforms, and large-scale scientific infrastructures. In such settings, centralizing raw data is often infeasible due to privacy, governance, or communication limitations. Federated Learning (FL) offers a way to tackle this challenge. However, most existing FL approaches are designed for machine learning problems and rely on assumptions such as independent local data that are fundamentally incompatible with spatial processes.

This talk introduces the concept of federated spatial modeling, which aims to perform statistically principled spatial inference across distributed data sources without sharing raw observations. We discuss the main requirements for federated spatial modeling, such as preserving spatial dependence across sites, supporting heterogeneous data and computational resources, enabling communication-efficient inference, and maintaining statistical interpretability. We claim that existing models and optimization strategies must be redesigned, rather than directly adapted from generic federated ML pipelines.

This talk highlights key challenges, design principles, and research directions for building scalable, robust, and privacy-aware spatial modeling frameworks suitable for real-world scientific and engineering applications.

A new isogeometric characteristic-Galerkin method based on the Non-Uniform Rational B-splines (NURBS) is developed in the current work to numerically solve the convection-dominated diffusion equations. The novel study incorporates the modified method of characteristics with the isogeometric analysis in a fractional-step framework in which the convective and the diffusive parts of the problem under study are treated separately. The proposed method maintains the advantages of the characteristic-Galerkin finite element method that time truncation errors are reduced and restrictions on the Courant numbers are alleviated in the simulations. In addition, the geometry of the computational domain is exactly presented by means of the NURBS basis functions. Unlike the conventional Eulerian techniques, the proposed isogeometric characteristic-Galerkin method is applied at each time step along the characteristic curves. Numerical results are presented for several test examples of convection-diffusion problems for which the exact solutions are available. The isogeometric characteristic-Galerkin method is also used for solving a nonlinear viscous Burgers problem. The obtained results demonstrate the ability of our new algorithm to accurately maintain the shape of the computed solutions in the presence of sharp gradients and shocks.

We present the lowest-order Neural Approximated Virtual Element Method (NAVEM) on polygonal meshes, a numerical approach that combines the Virtual Element Method (VEM) with neural networks to locally approximate virtual element basis functions. The key idea is to replace the implicit virtual basis functions with explicit neural-networkbased approximations constructed as linear combinations of harmonic functions. This strategy completely removes the need for polynomial projection operators and stabilization terms, which are among the most delicate and problem-dependent components of the standard VEM formulation.

The neural networks are not trained for individual mesh elements; instead, a single network is constructed for each class of polygonal elements sharing the same number of vertices and is then applied to all elements belonging to that class. This classification-based strategy allows the method to efficiently generalize across different meshes while keeping the offline training cost under control.

The proposed approach shifts the computational effort associated with the construction of the basis functions to an offline training phase, whereas the online phase closely resembles a classical finite element assembly on polygonal meshes. Building on the original NAVEM framework [3], we extend the method to general polygonal elements by enriching the local approximation space with additional harmonic functions, enabling an accurate representation of local singular behaviors near polygon vertices.

Extensive numerical experiments on general polygonal meshes, including triangular meshes with hanging nodes, confirm the effectiveness of the proposed method [1]. The results show that NAVEM preserves the expected convergence rates of the standard VEM while significantly reducing the difficulties related to stabilization parameter tuning and polynomial projections, and it exhibits improved robustness for strongly anisotropic and highly non-linear problems.

Bayesian inference on large-scale models is limited by computational feasibility, a trend that is further exacerbated by the continuous increase of data availability and model complexity. To address this issue, we present DALIA, a novel standalone Python implementation of the methodology of integrated nested Laplace approximations (INLA). Our framework aims at combining modern coding practices, code modularity and ease of use. It achieves unprecedented scalability through GPU-acceleration and a triple-layer, nested, distributed-memory parallelization strategy. We showcase its capabilities on a large-scale spatio-temporal multivariate air pollution study, scaling up to 496 Hopper GPUs on the CSCS Alps supercomputer. Our broader goal is to extend the scope of our novel framework to support other large-scale models that exceed the computational limits of the R-INLA package.

We apply a monotonicity-based numerical method to a time-dependent Mean Field Games (MFG) price formation model. This model is governed by a coupled system of Hamilton-Jacobi and transport equations and a market clearing condition. We design an iterative algorithm and test it in various problems with convex Hamiltonians. This method provides a framework for computing price dynamics in large-scale economic systems, with potential applications in market modeling.

We show that a broad range of convex optimization algorithms, including alternating projection, operator splitting, and multiplier methods, can be systematically derived from the framework of subspace correction methods via convex duality. To formalize this connection, we introduce the notion of dualization, a process that transforms an iterative method for the dual problem into an equivalent method for the primal problem. This concept establishes new connections across these algorithmic classes, encompassing both well-known and new methods. In particular, we show that classical algorithms such as the von Neumann, Dykstra, Peaceman–Rachford, and Douglas–Rachford methods can be interpreted as dualizations of subspace correction methods applied to appropriate dual formulations. Beyond unifying existing methods, our framework enables the systematic development of new algorithms for convex optimization. For instance, we derive parallel variants of alternating projection and operator splitting methods, as dualizations of parallel subspace correction methods, that are well-suited for large-scale problems on modern computing architectures and offer straightforward convergence guarantees. We also propose new alternating direction method of multipliers-type algorithms, derived as dualizations of certain operator splitting methods. These algorithms naturally ensure convergence even in the multi-block setting, where the conventional method does not guarantee convergence when applied to more than two blocks. This unified perspective not only facilitates algorithm design and the transfer of theoretical results but also opens new avenues for research and innovation in convex optimization.

Training deep neural networks (NNs) gives rise to large-scale, highly nonconvex optimization problems, making optimizer performance sensitive to hyperparameter choice. We present the Additively Preconditioned Trust-Region Strategy (APTS), which couples two ideas central to mathematical and computational data science: nonlinear domain-decomposition preconditioning (via parallel subdomain solves) and trust-region (TR) globalization.

Our approach partitions the network parameter vector into subdomains assigned to parallel devices. Each preconditioner iteration solves subdomain optimization problems in parallel to produce local nonlinear corrections, which are then combined additively into a global preconditioned trial step. A subsequent global TR step evaluates the full objective and updates the radius, providing stable progress and mitigating step-size sensitivity.

Building on APTS, we also introduce a non-monotone variant with a windowed acceptance rule that increases step acceptance and lowers wasted computation by permitting controlled non-monotonicity over a recent window.

Over its 38-year history, the Gordon Bell Prize has strongly influenced the development of high performance simulation and big data statistics in mostly positive, but also in some negative, ways, with implications for the operations of leading supercomputer centers. We briefly discuss the evolution of the prize in its several awarded categories then focus on a handful of submissions of influence. For example, driven by insights from Bill Gropp, the 1999 Gordon Bell Special Prize paper that first addressed an unstructured grid PDE application is presented as a pre-cursor of the 2009 roofline model now universally used to assess whether an application is compute- or bandwidth-bound, or even instruction-bound at a finer examination. The flop/s rate orientation of the Prize has encouraged wasteful flops, where a smaller number would provide more cost-effective delivery of quantities of scientific interest to working accuracy; we provide examples from four of our own recent submissions. How to pilot the Prize at the confluence of the roaring rivers of simulation and learning, with the tributary of quantum computing about to join, is the final topic of discussion. As energy costs now dominate what can be simulated and what can be learned or inferred, the elusive metric of "science per Joule" is proposed as one of several aspects to be addressed in the future evolution of this inspiring legacy.

We develop a systematic theory for the approximation capabilities and spectral behavior of shallow ReLU neural networks. Our recent results on linearized neural networks, in which the inner parameters are fixed, show that such models can achieve approximation rates comparable to those of fully nonlinear shallow ReLU networks, provided the target function lies in a slightly smaller class—namely, a Sobolev space rather than the Barron space in nonlinear approximation. We further investigate the metric entropy of the classes, demonstrating that their expressive complexity is comparable to that of high-order Sobolev classes. This analysis indicates that neural networks do not, in fact, fundamentally overcome the curse of dimensionality.

We further establish a saturation phenomenon for shallow ReLU models and conduct a detailed spectral analysis of the associated mass and stiffness matrices, providing insight into conditioning and stability when these models are used as discretization schemes for partial differential equations. Numerical experiments validate the theoretical predictions and illustrate that linearized networks offer a competitive alternative to traditional finite element methods for PDE approximation.

Understanding and quantifying complex interactions between oceanographic variables and marine ecosystems remains a major challenge, particularly in the context of climate change. Traditional large-scale ocean models often struggle to capture nonlinear, multiscale dependencies that govern biological responses in heterogeneous environments.

In the present work, we present a copula-based statistical framework designed to model complex dependence structures between sea surface temperature (SST) and chlorophyll-a concentration, allowing for a flexible representation of nonlinear and non-Gaussian relationships. This approach provides an effective alternative to conventional modeling strategies and enables a more detailed assessment of climate-driven impacts on primary productivity across diverse marine environments.

The framework is further extended to investigate the relationships between fish stock variability and multiple environmental indices, where copulas are used to disentangle joint dependencies that cannot be captured through standard correlation-based analyses. Building on this formulation, we introduce a copula-based sensitivity analysis, which allows the identification of key environmental drivers whose variability predominantly controls changes in fish stocks. This analysis is performed for several pelagic species and across contrasting environmental settings.

All experiments are conducted over the Strait of Gibraltar and along the Moroccan Mediterranean and Atlantic coastlines, regions characterized by strong hydrodynamic gradients and high ecological variability. The results highlight the ability of copula-based methods to reveal dominant climate–ecosystem linkages and to provide robust insights into the response of marine biological systems under changing climatic conditions.

We investigate the dimension dependence of Bramble–Pasciak–Xu (BPX) preconditioners for high-dimensional partial differential equations and show that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires deriving the dimension dependence of several fundamental tools in the theory of finite element methods, including elliptic regularity, the Bramble–Hilbert lemma, trace inequalities, inverse inequalities, and the Scott–Zhang interpolation. These results have important implications for emerging quantum computing methodologies: recent studies suggest that polynomial dimension dependence of BPX preconditioners may enable exponential speedups for quantum algorithms over their classical counterparts. This work is joint with Boou Jiang (KAUST) and Jinchao Xu (KAUST).

Predicting cancer cell proliferation is a fundamental challenge in mathematical oncology, traditionally addressed through mechanistic ordinary differential equations (ODEs). While classical models such as the Gompertz and Logistic growth equations offer high interpretability, they often struggle to assimilate sparse, noisy data from in vitro experiments.

This research presents a Scientific Machine Learning (SciML) framework that evolves from traditional numerical modeling to Physics-Informed Neural Networks (PINNs) and Neural Operators. Utilizing classical ODEs as a rigorous benchmark, we first employ PINNs to solve the inverse problem, identifying latent growth and diffusion parameters within a mechanistic context directly from primary experimental data. This data, representing cell counts from in vitro studies at Jarvis Christian University, allows us to validate the PINN's ability to maintain physical consistency where traditional numerical methods may fail due to experimental noise.

Expanding beyond instance-specific solvers, we introduce an Operator Learning approach (e.g., DeepONets) to learn the solution operator mapping treatment initial conditions to resulting growth trajectories. We provide a comparative analysis between traditional ODE benchmarks, PINNs, and Neural Operators, specifically addressing the "small-data" regime common in biological laboratory settings.

Supported by an NSF CISE-MSI Grant, this interdisciplinary collaboration between the Mathematics and Biology departments demonstrates how embedding classical growth laws into neural architectures enhances generalization across diverse experimental cohorts. This work advances the use of data-driven operators as robust, real-time digital twins for predictive mathematical oncology.

In the present contribution, we develop and assess a Proper Orthogonal Decomposition–Polynomial Chaos Expansion (POD–PCE) surrogate modelling framework for uncertainty propagation and quantification in hydrodynamic simulations. The underlying physical model is based on a system of multilayer shallow water equations that account for interlayer mass exchange and stochastic bottom topography. For the numerical solution, we employ a finite volume characteristic-based method that avoids the explicit use of the system's eigenstructure, resulting in a solver that is both computationally efficient and accurate for simulating slow and fast hydraulic regimes.

The proposed framework enables the systematic analysis of the propagation and impact of multiple uncertain parameters in multilayer shallow water flow models. To alleviate the high computational cost typically associated with uncertainty quantification, we integrate Proper Orthogonal Decomposition with Polynomial Chaos Expansion, significantly reducing the number of model evaluations required in the presence of high-dimensional random inputs.

The effectiveness of the approach is demonstrated through several numerical experiments, including wind-driven recirculation flows over flat and variable bathymetries. In addition, a realistic case study involving recirculation flows in the Strait of Gibraltar is presented. The results highlight the robustness and accuracy of the proposed POD–PCE surrogate model when compared to standard Monte Carlo methods, while achieving substantial computational savings. These findings indicate that surrogate-based uncertainty quantification provides an efficient and reliable alternative for complex hydrodynamic applications.

Spatial data are central to applications such as environmental monitoring and urban planning, but are often distributed across devices where privacy and communication constraints limit direct sharing. Federated modeling offers a practical solution that preserves data privacy while enabling global modeling across distributed data sources. For instance, environmental sensor networks are privacy- and bandwidth-constrained, motivating federated spatial modeling that shares only privacy-preserving summaries to produce timely, high-resolution pollution maps without centralizing raw data. However, existing federated modeling approaches either ignore spatial dependence or rely on synchronous updates that suffer from stragglers in heterogeneous environments. This work proposes an asynchronous federated modeling framework for spatial data based on low-rank Gaussian process approximations. The method employs block-wise optimization and introduces strategies for gradient correction, adaptive aggregation, and stabilized updates. We establish linear convergence with explicit dependence on staleness, a result of standalone theoretical significance. Moreover, numerical experiments demonstrate that the asynchronous algorithm achieves synchronous performance under balanced resource allocation and significantly outperforms it in heterogeneous settings, showcasing superior robustness and scalability.

The cyclic stepsize update strategy is proposed for stochastic gradient method. The stepsize is updated cyclicly, where the first two stepsizes use the approxiamted Cauchy steps and the rest apply the fixed stepsize. The step-ahead BB stepsize is used for the Cauchy step approximation. The method combines with both monotone and nonmonotone linesearches. The convergence properties are analyzed under different types of problems, where the theoretical results are proved without the interpolation condition assumption. The numerical tests show good performances of the proposed methods compared to other first order stochastic methods.

We introduce conditional push-forward neural networks (CPFN), a generative framework for conditional distribution estimation. Instead of directly modeling the conditional density f_Y|X, CPFN learns a stochastic map φ=φ(x,u) such that φ(x,U) and Y|X=x follow approximately the same law, with U a suitable random vector of pre-defined latent variables. This enables efficient conditional sampling and straightforward estimation of conditional statistics through Monte Carlo methods. The model is trained via an objective function derived from a Kullback–Leibler formulation, without requiring invertibility or adversarial training. We establish a near-asymptotic consistency result and demonstrate experimentally that CPFN can achieve performance competitive with—or even superior to—state-of-the-art methods, including kernel estimators, tree-based algorithms, and popular deep learning techniques, all while remaining lightweight and easy to train.

Monotone operator theory provides a standard method for proving existence of solutions to boundary value problems involving elliptic PDEs, generalizing both the direct method in the calculus of variations and the Lax-Milgram theorem. Mean field games (MFG) possess a similar monotonicity structure, recognized since Lasry and Lions (2007). Cardaliaguet (2015) established existence of weak solutions to first-order, local, separable MFG systems via variational methods. Ferreira, Gomes, and collaborators (2018, 2019, 2021) formulated more general MFG systems as monotone operator variational inequalities and proved these admit solutions, though the connection to Cardaliaguet-type weak solutions remained incomplete. In this talk, I present recent joint work with Ferreira and Gomes that bridges this gap for stationary, first-order, local mean field games. We give a streamlined proof that the variational inequality admits a solution under assumptions broad enough to include non-separable systems and soft congestion models, and we establish that this solution is a weak solution in Cardaliaguet's sense. This unifies the monotone operator and variational approaches while extending rigorous existence results to a broader class of MFG systems relevant for pedestrian dynamics and traffic flow.

We will present several stabilized finite element methods for advection–diffusion equations of different differential forms, with particular emphasis on recent extensions to vector-valued problems. We introduce a class of unwinding-type schemes, together with exponentially fitted methods, and also present a robust multigrid solver for H(curl) convection–diffusion problems. These developments illustrate concrete strategies for the design and analysis of discretizations for general convection–diffusion problems, and demonstrate how classical scalar techniques can be systematically generalized to accommodate the distinctive mathematical structure of vector field spaces.

This talk presents a unified framework linking Barron and Sobolev spaces to analyze the approximation properties of neural networks with ReLU-type activation functions. Within this framework, we establish both classical and new sharp approximation results.

We show that for functions in an appropriate Barron space, ReLU networks can achieve high approximation accuracy without suffering from the curse of dimensionality. The same approximation rate is obtained in a corresponding Sobolev space when using linearized ReLU networks, and these results are compared with those for classical global polynomial approximations.

A key observation is that the relevant Barron and Sobolev spaces have comparable complexity, as measured by metric entropy. As a consequence, a piecewise linear finite element space expressed in terms of ReLU networks can avoid the curse of dimensionality for sufficiently smooth functions, whereas the classical linear finite element space cannot.

Finally, we introduce a bit-centric perspective showing that parameter count alone is not a reliable measure of approximation complexity. Together, these results clarify the mathematical connections between finite element analysis and deep learning, and provide insights into scientific machine learning.

We prove a weak convergence result for a single forward-step projective splitting method for a finite sum of monotone inclusion problems without cocoercivity. Our result extends and complements several existing results in the literature. We apply the obtained result to proton treatment planning problems arising in radiotherapy optimization for cancer treatment.

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for analyzing data produced by computational science and engineering applications using deep-learning techniques. At the core of these methods is the supervised training algorithm to learn the neural network realization, a highly non-convex optimization problem that is usually solved using stochastic gradient methods.

However, distinct from deep-learning practice, scientific machine learning training problems feature a much larger volume of smooth data and better characterizations of the empirical risk functions, which make them suited for conventional solvers for unconstrained optimization.

In this talk, we introduce PETScML, a lightweight software framework built on top of the Portable and Extensible Toolkit for Scientific computation (PETSc) to bridge the gap between deep-learning software and conventional solvers for unconstrained minimization.

Using PETScML, we empirically demonstrate the superior efficacy of a trust-region method based on the Gauss-Newton approximation of the Hessian in improving generalization errors in regression tasks when learning surrogate models across a wide range of scientific machine-learning techniques and test cases. All the conventional solvers tested, including L-BFGS and inexact Newton with line search, compare favorably with the adaptive first-order methods used to validate the surrogate models in terms of both cost and accuracy.

We build AraLLaMA, a large language model for Arabic, addressing the limited support for Arabic compared to mainstream languages such as English and Chinese. To improve decoding efficiency while avoiding knowledge degradation caused by out-of-vocabulary issues, we introduce Progressive Vocabulary Expansion, inspired by second language acquisition. Our experiments show that this approach is effective, and AraLLaMA achieves competitive performance across multiple Arabic benchmarks.