In this talk, I will present a series of recent works that explore the intersection of deep learning and scientific computing, with a particular focus on operator learning and generative modeling for PDEs. I will begin with our work on Convolutional Neural Operators (CNOs), which reinterprets operator learning through the lens of classical CNNs. I will then discuss Representation Equivalent Neural Operators (ReNO), where we address representational ambiguities and aliasing in operator learning. Building on these foundations, I will introduce Poseidon, a foundation model for time-dependent PDEs that leverages multiscale attention and temporal conditioning. Finally, I will present GenCFD, a diffusion-based generative model for statistical CFD, which captures uncertainty and high-order statistics in turbulent flows. Each of these works contributes a step toward building accurate, robust, and generalizable models for physical systems.

We discuss a Bayesian framework for learning non-linear mappings in infinite-dimensional spaces. Given a map $g_0:\mathcal{X}\to\mathcal{Y}$ between two separable Hilbert spaces, we study recovery of $g_0$ from $n\in\mathbb{N}$ noisy input-output pairs $(\pmb{x},\pmb{y})=(X_i,Y_i)_{i=1}^n$ with $Y_i = g_0 (X_i ) + E_i$ ; here the $X_i\in\mathcal{X}$ represent randomly drawn 'design' points, and the $E_i$ are assumed to be i.i.d. draws from a Gaussian white noise process indexed by $\mathcal{Y}$. Choosing a suitable 'operator-valued' prior $\Pi_0$, we show well-posedness of the posterior $G|(\pmb{X},\pmb{Y})$ and establish convergence rates for the posterior mean towards the ground truth in terms of the data size $n$.

Modern decision-making for complex physical and engineered systems increasingly requires the ability to quantify high-dimensional uncertainties and make decisions by solving risk-averse optimization problems under uncertainty all in near real-time. Operator learning has emerged as a promising framework for enabling scalable surrogate modeling in this context. However, such approaches inevitably introduce approximation errors that can degrade the quality of downstream decisions. In this talk, we explore how formulating operator learning tasks through the lens of decision-making objectives such as inverse problems and optimization under uncertainty can help mitigate these limitations. We derive a priori error bounds for these optimization tasks, which motivate operator learning formulations that explicitly penalize errors in the derivatives of input-output maps. We demonstrate the effectiveness of these methods on problems in structural health monitoring, shape optimization, and fluid flow control. Our approaches lead to (i) improved accuracy in surrogate-based optimization and (ii) empirically enhanced performance in statistical learning tasks.

This work introduces an interpretable graph neural network (GNN) autoencoding framework for surrogate modeling of unstructured mesh fluid dynamics. The approach addresses two key challenges: latent space interpretability and compatibility with unstructured meshes. Interpretability is achieved through an adaptive graph reduction procedure that performs flowfield-conditioned node sampling, yielding masked fields that (i) localize latent activity in physical space, (ii) track the evolution of unsteady flow features such as recirculation zones and shear layers, and (iii) identify sub-graphs most relevant to forecasting. Compatibility with unstructured meshes is ensured through multi-scale message passing (MMP) layers, which extend standard message passing with learnable coarsening operations to efficiently reconstruct flowfields across multiple lengthscales. A regularization procedure further enhances interpretability by enabling error-tagging, i.e., the identification of nodes most associated with forecasting uncertainty. Demonstrations are conducted on large-eddy simulation data of backward-facing step flows at high Reynolds numbers, with extrapolations to ramp and wall-mounted cube configurations. The result is a new class of GNN autoencoders that combine predictive accuracy with physics-informed insight into both flow structures and model reliability.

Physics-informed neural networks (PINNs) face significant challenges when solving differential equations with rapid oscillations, steep gradients, or singular behavior. Considering these challenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to address this class of differential equations. We represent the solution in wavelet space using a family of smooth-compactly supported wavelets. This approach captures the dynamics of complex physical phenomena with significantly fewer degrees of freedom, enabling faster and more accurate training by searching for solutions within the wavelet space. Developed model eliminates the need for automatic differentiation of derivatives in differential equations and requires no prior knowledge about solution behavior, such as the location of abrupt features. In this talk, I will demonstrate how W-PINNs excel at capturing localized nonlinear information through a strategic fusion of wavelets with PINNs, making them particularly effective for problems exhibiting abrupt behavior in specific regions. Our experimental findings show that W-PINNs significantly outperform traditional PINNs, PINNs with wavelets as an activation function, and other state-of-the-art methods, offering a promising approach for tackling challenging differential equations in scientific and engineering applications.

Mathematically, operator surrogates provide approximations of mappings between infinite dimensional spaces. We analyze an encoder-decoder framework, where the input and output function spaces are parametrized using coefficient sequences of admissable representation systems. The goal then becomes to approximate a coefficient-to-coefficient map, which can be realized using various approximation tools. In the present work we focus on low-rank tensor representation using the tensor-train (TT) format. This format allows for efficient storage, evaluation and basic linear algebra as long as the tensor ranks remain low. We show approximation rates with regards to the storage complexity of such TT surrogates for certain holomorphic maps by providing rank bounds. Furthermore, we draw comparisons to other approximation tools such as neural networks and sparse polynomial interpolation.

Deep autoencoders have become a fundamental tool in various machine learning applications, ranging from dimensionality reduction and reduced order modeling of partial differential equations to anomaly detection and neural machine translation. However, despite their empirical success, the theoretical understanding of these architectures remains relatively underdeveloped, particularly in contrast to classical linear approaches such as Principal Component Analysis. In this talk, I will overview some of the theory concerning deep autoencoders, addressing different aspects through different mathematical tools. More precisely, I will discuss: (i) the relationship between the latent dimension of deep autoencoders and some results in topology and dimension theory, such as the Borsuk-Ulam theorem and the Menger-Nöbeling theorem; (ii) the extension of such results to the context of stochastic systems, where Gaussian processes enter into play; (iii) some results about the expressivity of deep autoencoders, specifically focusing on convolutional architectures and, finally, (iv) some recent advancements concerning the design and optimization of symmetric architectures inspired by the Eckart-Young theorem. Theoretical results will be accompanied by suitable numerical experiments, illustrating the interplay between theory and practice.

Many natural and engineered systems remember their past, for example, materials that slowly deform, fluids with long-lasting vortices, or biological populations affected by historical conditions. Traditional models often struggle to capture this memory effect, but fractional differential equations provide a powerful mathematical framework. In this talk, I will introduce fractional dynamics using simple examples to show how past states influence present behaviour. Using visualizations and simulations, I will demonstrate how learning-based methods, such as neural networks, can efficiently approximate these systems. While the focus is on conceptual understanding, I will also briefly highlight how these methods connect to advanced research areas, such as operator learning, neural PDE solvers, and physics-informed learning. The goal is to provide a clear understanding of memory-dependent systems while presenting a path to more advanced modelling techniques.

Filtering - the task of estimating the conditional distribution for states of a dynamical system given partial and noisy observations - is important in many areas of science and engineering, including weather and climate prediction. However, the filtering distribution is generally intractable to obtain for high-dimensional, nonlinear systems. Filters used in practice, such as the ensemble Kalman filter (EnKF), provide biased probabilistic estimates for nonlinear systems and have numerous tuning parameters. I will present a framework for learning a parameterised analysis map - the transformation that takes samples from a forecast distribution, and combines with an observation, to update the approximate filtering distribution - using variational inference. In principle this can lead to a better approximation of the filtering distribution, and hence smaller bias. We show that this methodology can be used to learn the gain matrix, in an affine analysis map, for filtering linear and nonlinear dynamical systems; we also study the learning of inflation and localisation parameters for an EnKF. The framework developed here can also be used to learn new filtering algorithms with more general forms for the analysis map. I will also present some recent work on learning corrections to the EnKF using permutation-invariant neural architectures, leading to superior performance compared to leading methods in filtering chaotic systems.