𝝅 - Seminar Series

The virtual Param-Intelligence (𝝅) seminar series aims to provide a dynamic platform for researchers, engineers, and students to explore and discuss the latest advancements in integrating machine learning with scientific computing. Key topics include data-driven modeling, physics-informed neural surrogates, neural operators, hybrid computational methods, and novel machine/deep learning algorithms with a strong focus on real-world applications across various fields of computational science and engineering.

Speakers

• Dr. Jan Drgona, Pacific Northwest National Laboratory (PNNL), USA,  October 24, 2024, (12 - 1 pm ET). [YouTube Video]

Title: Differentiable Programming for Data-driven Modeling, Optimization, and Control

Abstract: This talk will present a different programming perspective on physics-informed machine learning (PIML). Specifically, we will discuss the opportunity to develop a unified PIML framework for digital twins of dynamical systems, learning to optimize, and learning to control methods. We demonstrate the performance of these emerging PIML methods in a range of engineering case studies, including modeling of networked dynamical systems, robotics, building control, and dynamic economic dispatch problem in power systems.

• Prof. Romit Maulik, Pennsylvania State University, USA, November 07, 2024, (12 - 1 pm ET). [YouTube Video]

Title: Neural ordinary differential equations for scientific machine learning

Abstract: In this talk, I will discuss recent advancements in the construction and training of neural ordinary differential equations (Neural ODEs) for learning complex dynamical systems characterized by chaotic and multiscale behavior. In particular, my talk will discuss the challenges associated with learning invariant statistics of dynamical systems given solely short-term predictive performance based objective functions. We will introduce neural architectures and algorithms in the neural ODE paradigm that leverage partial knowledge of the underlying dynamics to obtain surrogate models that can recover correct chaotic dynamics for a wide range of systems ranging from the 1D Kuramoto-Sivashinsky equations to turbulent Navier-Stokes equations and the planetary atmosphere and ocean systems.

• Prof. Ahmed. H. Qureshi, Purdue University, USA, November 14, 2024, (12 - 1 pm ET). [YouTube Video]

Title: Neural PDEs for Robot Mapping and Motion Planning

Abstract: Robot motion planning involves finding a collision-free path between a robot's initial and target configurations. Recently, imitation learning-based methods have shown promise in efficiently generating paths at runtime using function approximators. However, these methods rely on a significant amount of expert trajectories for learning, which can be computationally expensive to produce. Therefore, this presentation will discuss a new class of self-supervised, physics-informed neural motion planners. These methods directly learn to solve the Eikonal Partial Differential Equation (PDE) for motion planning without relying on expert data for learning. Additionally, the talk will also cover how solving Eikonal PDE leads to a new robot mapping feature called the arrival time fields, which can be inferred online to map an unknown environment via exploration. This new mapping feature is better suited for motion planning than existing mapping features such as occupancy maps or Sign Distance Fields, allowing extremely fast inference of path solutions in the mapped environment. The results also demonstrate that these new approaches outperform state-of-the-art traditional and imitation learning-based motion planning methods in terms of computational planning speed, path quality, and success rates.

• Prof. Matthias Möller, Delft University of Technology, Netherlands, November 21, 2024, (12 - 1 pm ET). [YouTube Video]

Title:  Bridging the gap between isogeometric analysis and deep operator learning

Abstract: Isogeometric Analysis (IgA) introduced by Hughes et al. in 2005 has revived the vision of design-through-analysis (DTA) originally proposed by Augustitus et al. in 1977. DTA means the fully virtual creation, analysis and optimization of engineering designs, which requires bidirectional exchange of data between computer-aided design (CAD) and engineering analysis (CAE) tools. While IgA targets at bridging the gap between CAD and CAE through the use of spline-type basis functions throughout the entire process, the full potential of DTA is hold back by high computational costs of simulation-based analysis tools that hinder truly interactive DTA workflows. In this presentation we will briefly review the mathematical basics of IgA and present a novel approach – IgANets – that integrates the concept of deep operator learning into the isogeometric framework. In particular, we show that IgANets can be interpreted as a network-based variant of least-squares collocation IgA (Lin et al. 2020), thereby inheriting its consistency and convergence properties. We will moreover present a software prototype that enables the collaborative creation and analysis of designs across multiple end-user devices including tablets and VR/XR headsets.

• Prof. Marta D'Elia, Stanford University, USA, December 05, 2024, (12 - 1 pm ET). [YouTube Video]

Title:  On the use of Graph and Point networks in scientific applications

Abstract: In the context of scientific and industrial applications, one often has to deal with unstructured space-time data obtained from numerical simulations. The data can be either in the form of a mesh or a point cloud. In this context, graph neural networks (GNNs) have proved to be effective tools to reproduce the behavior of simulated data; however, depending on the physical nature of the datasets, variations of vanilla GNNs have to be considered to ensure accurate results. Furthermore, when only a point cloud is available, one can also consider a graph-free approach by building a "point network" that doesn't require connectivity information. In this presentation we focus on particle-accelerator simulations; a computationally demanding class of problems for which rapid design and real-time control are challenging. We propose a machine learning-based surrogate model that leverages both graph and point networks to predict particle-accelerator behavior across different machine settings. Our model is trained on high-fidelity simulations of electron beam acceleration, capturing complex, nonlinear interactions among macroparticles distributed across several initial state dimensions and machine parameters. Our initial results show the model’s capacity for accurate, one-shot tracking of electron beams at downstream observation points, outperforming baseline graph convolutional networks. This framework accommodates key symmetries inherent in particle distributions, enhancing stability and interpretability. We also mention our ongoing work focused on extending these methods to autoregressive tracking across multiple timesteps. This research offers a powerful approach to reducing computational demands in particle-accelerator simulations, contributing to advancements in real-time optimization and control. 

• Dr. Christopher Rackauckas, Massachusetts Institute of Technology, USA, December 12, 2024, (12 - 1 pm ET). [YouTube Video]

Title: Enabling Industrially-Robust AI for Engineering through Scientific Machine Learning

Abstract: The combination of scientific models into deep learning structures, commonly referred to as scientific machine learning (SciML), has made great strides in the last few years in incorporating models such as ODEs and PDEs into deep learning through differentiable simulation. Such SciML methods have been gaining steam due to accelerating the development of high-fidelity models for improving industrial simulation and design. However, many of the methods from the machine learning world lack the robustness required for scaling to industrial tasks. What needs to change about AI in order to allow for methods which can guarantee accuracy and quantify uncertainty? In this talk we will go through the details of how one can enable robustness in building and training SciML models. Numerical robustness of algorithms for handling neural networks with stiff dynamics, continuous machine learning methods with certifiably globally-optimal training, alternative loss functions to mitigating local minima, integration of Bayesian estimation with model discovery, and tools for validating the correctness of surrogate models will be discussed to demonstrate a next generation of SciML methods for industrial use. Demonstrations of these methods in applications such as two-phase flow HVAC systems, modeling of sensors in Formula One cars, and lithium-ion battery packs will be used to showcase the improved robustness of these approaches over standard (scientific) machine learning.

• Dr. Shuvayan Brahmachary, Shell Inc.,India, December 19, 2024, (12 - 1 pm ET). [YouTube Video]

Title:  Large language models as evolutionary optimizers for industrial problems

Abstract:  Large Language Models (LLMs) have shown impressive capabilities in reasoning and problem-solving, making them intriguing candidates for optimization tasks. This work explores the potential of LLMs as zero-shot optimizers across various complex scenarios, including multi-objective and high-dimensional problems. A novel approach, termed the Language-Model-Based Evolutionary Optimizer (LEO), is introduced, leveraging the unique strengths of LLMs for numerical optimization. Through a series of examples, ranging from benchmark tests to real-world engineering challenges like supersonic nozzle design and windfarm layout, the effectiveness of LEO is demonstrated. Comparisons with traditional optimization methods highlight the competitive performance of LLMs, while also addressing the challenges posed by their creative tendencies. Practical insights and future research directions are discussed to harness the full potential of LLMs in optimization.

• Jassem Abbasi, Ph.D. Candidate, University of Stavanger, Norway, January 09, 2025, (12 - 1 pm ET). [YouTube Video]

Title:  History-Matching of Imbibition Flow in Multiscale Fractured Porous Media Using Physics-Informed Neural Networks (PINNs)

Abstract:  In this talk, I am going to present our recent work on utilization of physics-informed neural networks (PINNs) for inverse calculation of dynamic properties of multiphase flow in porous media, by history matching a multi-fidelity experimental dataset. After validating the workflow in forward and inverse modeling of a synthetic problem of flow in fractured porous media, we applied it to a real experimental dataset in which brine is injected at a constant pressure drop into a CO2 saturated naturally fractured shale core plug. The exact spatial positions of natural fractures and the dynamic in-situ distribution of fluids were imaged using a CT-scan setup. To model the targeted system, we followed a domain decomposition approach for matrix and fractures and a multi-network architecture for the separate calculation of water saturation and pressure. The flow equations in the matrix, fractures and interplay between them were solved during training. Prior to fully coupled simulations, we suggested pre-training the model. This aided in a more efficient and successful training of the coupled system. Both for the synthetic and experimental inverse problems, we determined flow parameters within the matrix and the fractures. Multiple random initializations of network and system parameters were performed to assess the uncertainty and uniqueness of the resulting calculations. The results confirmed the precision of the inverse calculated parameters in retrieving the main flow characteristics of the system. The consideration of multiscale matrix-fracture impacts is commonly overlooked in existing workflows. Accounting for them led to several orders of magnitude variations in the calculated flow properties compared to not accounting for them. To the best of our knowledge, the proposed PINNs-based workflow is the first to offer a reliable and computationally efficient solution for inverse modeling of multiphase flow in fractured porous media, achieved through history-matching noisy and multi-fidelity experimental measurements.

• Dr. Youngsoo Choi, Lawrence Livermore National Laboratory (LLNL), USA, January 16, 2025, (12 - 1 pm ET). [YouTube Video]

Title:  LaSDI: Latent space dynamics identification

Abstract:  Many computational models for physical systems have been developed to expedite scientific discovery, which would otherwise be impeded by the lengthy nature of traditional, non-computational experimentation (e.g., observation, problem identification, hypothesis formulation, experimentation, data collection, analysis, conclusion, and theory development). However, as these physical systems grow more complex, the computational models themselves can become prohibitively time-consuming. To address this challenge, we introduce a framework called Latent Space Dynamics Identification (LaSDI), which transforms complex, high-dimensional computational domains into reduced, succinct coordinate systems—a sophisticated change of variables that preserves essential dynamics. LaSDI offers significant potential for extension to other innovative, data-driven algorithms. It is an interpretable, data-driven framework composed of three core steps: compression, dynamics identification, and prediction. In the compression phase, high-dimensional data is reduced to a more manageable form, facilitating the construction of an interpretable model. The dynamics identificationphase then derives a model, typically expressed as parameterized differential equations, that accurately captures the behavior of the reduced latent space. Finally, in the prediction phase, these differential equations are solved within the reduced space for new parameter sets, with the resulting solutions projected back into the full space. One of the key advantages of LaSDI is its computational efficiency, as the prediction phase operates entirely in the reduced space, bypassing the need for the full-order model. The LaSDI framework supports various identification methods, including fixed forms like dynamic mode decomposition and thermodynamics-based LaSDI, regression methods such as sparse identification of nonlinear dynamics(SINDy) and weak SINDy, as well as physics-driven approaches like projection-based reduced order models. The LaSDI family has demonstrated substantial success in accelerating various physics problems, achieving up to 1000x speed-ups in fields such as kinetic plasma simulations, pore collapse phenomena, and computational fluid dynamics

• Prof. Michael P. Brenner, Harvard University, USA, January 30, 2025, (12 - 1 pm ET). [YouTube Video]

Title:  Scientific Uses of Automatic Differentiation

Abstract: There is much excitement (some of it legitimate) about applications of machine learning to the sciences. Here I’m going to argue that a primary opportunity is not machine learning per se, but instead that the tools underlying the ML revolution yield significant opportunities for scientific discovery. Primary among these tools is automatic differentiation and the scalability of codes. Neural network architectures are similar to time rollouts in dynamical systems, and therefore the technical advances underlying the ML have the potential to directly translate into the ability to solve important optimization problems in the sciences that have heretofore not been tackled. I will describe a number of different directions we have been undertaking using automatic differentiation and large scale optimization to solve science problems, including developing new algorithms for solving partial differential equations, the design of energy landscapes and kinetic pathways for self assembly, the design of fluids with designer rheologies, “optimal porous media”, learning the division rules for models of tissue development, efficient algorithms for finding unstable periodic orbits in turbulent flows as high order descriptors of turbulent statistics and the development of neural general circulation models for weather and climate (where the physics parameterizations in the GCM are learned from fitting against data).  If I have time, I'll also touch on the fact that the new computational tools suggest entirely new ways of finding approximate theoretical descriptions of solutions of nonlinear PDEs, and will point to a linear approximation of the Navier Stokes equation that captures unsteady flow past a moving body up to Reynolds number of O(800).

• Prof. Amirhossein Arzani, University of Utah, USA, February 06, 2025, (12 - 1 pm ET). [YouTube Video]

Title:  XAI for science: from dynamical system discovery to interpretable operator learning

Abstract: Recent advances in scientific machine learning have demonstrated remarkable capabilities in modeling complex physical phenomena, yet challenges persist in balancing predictive power with interpretability and scientific understanding. In this talk, I will present two recent developments in our group related to explainable artificial intelligence (XAI) and scientific machine learning. First, I will present ADAM-SINDy, a novel extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that enables the discovery of nonlinear parameterized differential equations from observational data. By integrating the ADAM optimization algorithm with sparse regression techniques in a PyTorch environment, this method can simultaneously identify both governing equations and nonlinear parameters, which was not possible in the classical SINDy framework. Second, we will explore how functional data analysis principles can be leveraged to create interpretable operators in scientific applications. This approach not only provides analytical representations of learned operators in the form of integral equations but can also be used to understand spatial relationships between the data.

• Dr. Yongji Wang, Courant Institute, NYU, USA, February 13, 2025, (12 - 1 pm ET). [YouTube Video]

Title:  Multi-stage neural networks for multiscale dynamics

Abstract: Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. In this talk, I will use heuristic example to illustrate the principle and feature of the method. More examples are used to demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision of double-floating point within a finite number of iterations. This advancement mitigate the longstanding accuracy limitation of neural network training, and enhance PINNs as robust tools for solving challenging differential equations in mathematics, such as blow-up problems.

• Prof. Sergei Kalinin, University of Tennessee, Knoxville, USA, February 20, 2025, (11 - 12 pm ET). [YouTube Video]

Title:  Rewards are all we need: building autonomous materials synthesis and characterization workflows

Abstract: The trajectory of scientific research worldwide is guided by long-term goals, spanning the spectrum from curiosity and fundamental discoveries in physics to the applied challenges of enhancing materials and devices for a wide array of applications. However, the execution and assessment of daily research efforts typically hinges on multiobjective reward functions, which can be evaluated either during or at the conclusion of an experimental campaign. Although this concept is tacitly acknowledged within the scientific community, the implementation of autonomous experimental workflows in automated laboratories necessitates the formulation of robust reward functions and their seamless integration across various domains. Should these reward functions be universally established, the entirety of experimental efforts could be conceptualized as optimization problems. Here, I will present our latest advancements in the development of autonomous research systems based on electron and scanning probe microscopy, as well as for automated materials synthesis based on reward driven workflows and reward integration across domains. We identify several categories of reward functions that are discernible during the experimental process, including imaging optimization, fundamental physical discoveries, the elucidation of correlative structure-property relationships, and the optimization of microstructures. The operationalization of these rewards function on autonomous microscopes is demonstrated, as well as strategies for human in the loop intervention. Utilizing these classifications, we construct a framework that facilitates the integration of multiple optimization workflows, demonstrated through the synchronous orchestration of diverse characterization tools across a shared chemical space, and the concurrent navigation of costly experiments and models that adjust for epistemic uncertainties between them. 

• Dr. Donsub Rim, Washington University in St. Louis, USA, February 20, 2025, (12 - 1 pm ET).  [YouTube Video]

Title: FastLRNR and Sparse Physics Informed Backpropagation

Abstract: Many recent works are exploring the potential advantages in leveraging deep learning models in solving parametrized partial differential equations (pPDEs).  This talk concerns one approach leading to computational speed ups in the popular Physics Informed Neural Networks (PINNs) framework. We introduce specialized neural networks called Low Rank Neural Representations (LRNRs), whose weight and bias parameters are endowed with a singular-value-decomposition-like low rank structure. We show that backpropagation operations that are necessary in PINNs approaches can be computed efficiently due to the low rank structure. We call these efficient operations sparse physics informed backpropagation (SPInProp). SPInProp operations have computational complexity that scales only with a reduced dimension, which is independent of the width (resolution) of the neural network approximation. We present computational experiments that demonstrate LRNRs can be adapted to solve pPDEs using only SPInProps within PINNs. We will also discuss the approximation theoretical results that motivates the LRNR architecture, in connection to low dimensional representations of shock interactions.

• Dr. Emmanuel de Bézenac, INRIA, France, February 27, 2025, (12 - 1 pm ET) [YouTube Video]

Title: An operator preconditioning perspective on training in physics-informed machine learning

Abstract: We investigate the behavior of gradient descent algorithms in physics-informed machine learning methods, such as PINNs, which minimize residuals associated with partial differential equations (PDEs). Our analysis reveals that training difficulties are closely linked to the conditioning of a key differential operator, specifically the Hermitian square of the PDE's underlying operator. Poor conditioning of this operator leads to slow or ineffective optimization.

• Prof. Siddhartha Mishra, Department of Mathematics, ETH Zurich, Switzerland, March 06, 2025, (12 - 1 pm ET). [YouTube Video]

Title: Learning PDEs. 

Abstract: Despite their remarkable success over many decades, numerical methods for approximating PDEs can incur a very high computational cost. This limitation has provided impetus to the design of fast and accurate Machine Learning/AI based surrogates which can learn the PDE solution operator from data. In this talk, we review some latest developments in the field of Neural Operators, which are widely used as an ML paradigm for PDEs and discuss state of the art neural operators based on convolutions or attention. The issue of sample complexity is addressed by the design of general purpose Foundation models for PDEs. If time permits, we will also discuss graph based architectures for PDEs on arbitrary domains and conditional Diffusion models for PDEs with multiscale solutions. 

• Prof. Thomas Dietterich, Oregon State University, USA, March 13, 2025, (12 - 1 pm ET). 

Title: Can we make machine learning safe for safety-critical systems?

Abstract: The impressive new capabilities of systems created using deep learning are encouraging engineers to apply these techniques in safety-critical applications such as medicine, aeronautics, and self-driving cars. This talk will discuss the ways that machine learning methodologies are changing to operate in safety-critical systems. These changes include (a) building high-fidelity simulators for the domain, (b) adversarial collection of training data to ensure coverage of the so-called Operational Design Domain (ODD) and, specifically, the hazardous regions within the ODD, (c) methods for verifying that the fitted models generalize well, and (d) methods for estimating the probability of harms in normal operation. There are many research challenges to achieving these. But we must do more, because traditional safety engineering only addresses the known hazards. We must design our systems to detect novel hazards as well. We adopt Leveson’s view of safety as an ongoing hierarchical control problem in which controls are put in place to stabilize the system against disturbances. Disturbances include novel hazards but also management changes such as budget cuts, staff turnover, novel regulations, and so on. Traditionally, it has been the human operators and managers who have provided these stabilizing controls. Are there ways that AI methods, such as novelty detection, near-miss detection, diagnosis and repair, can be applied to help the human organization manage these disturbances and maintain system safety?

• Dr. Thomas O'Leary-Roseberry, University of Texas at Austin, USA, March 20, 2025, (12 - 1 pm ET). 

Title: Derivative-Informed Neural Operators

Abstract: Decision-making for complex physical systems is a core aspect of modern engineering. By employing computational models, engineers can explore many questions related to system design and control. However, because observations and computational models both contain inherent errors and uncertainties, robust statistical models must be developed. Such models enable (i) the statistical calibration of computational models to noisy observational data, and (ii) the formulation of risk-averse optimization strategies for system management. A major computational challenge arises because these statistical models often require repeated evaluations of expensive computational models for various input parameters. In this talk, we present ways to construct advanced machine learning approximations of high-fidelity computational models by leveraging the derivative information of model outputs with respect to their inputs. Our approaches (1) use this derivative information to identify low-dimensional subspaces that yield efficient dimension-reduced surrogate models, and (2) incorporate derivatives into training to produce accurate surrogates for derivative-based inference and optimization. These derivative-informed neural operators (DINOs) yield efficient, scalable, and state-of-the-art algorithms for Bayesian inverse problems and for optimal design and control of complex systems. Our methods are supported by rigorous analyses, including a priori error bounds for optimization, and demonstrate exceptional performance in numerical experiments. We demonstrate the effectiveness of our approaches on different neural operator architectures, including reduced-basis architectures and the Fourier Neural Operator. We also showcase the advantages of DINOs in a variety of statistical decision-making tasks, including challenging and query-intensive problems such as deterministic and Bayesian inverse problems, as well as PDE-constrained optimization under uncertainty. Numerical experiments include inferring material properties in elastic solids and controlling viscous flow around bluff bodies.

• Prof. Ricardo Vinuesa, KTH Royal Institute of Technology, Sweden, March 27, 2025, (11 am - 12 pm ET). 

Title: Turbulence control through explainable deep learning

Abstract: In this work we first use explainable deep learning based on Shapley explanations to identify the most important regions for predicting the future states of a turbulent channel flow. The explainability framework (based on gradient SHAP) is applied to each grid point in the domain, and through percolation analysis we identify coherent flow regions of high importance. These regions have around 70% overlap with the intense Reynolds-stress (Q) events in two-dimensional vertical planes. Interestingly, these importance-based structures have high overlap with classical turbulence structures (Q events, streaks and vortex clusters) in different wall-normal locations, suggesting that this new framework provides a more comprehensive way to study turbulence. We also discuss the application of deep reinforcement learning (DRL) to discover active-flow-control strategies for turbulent flows, including turbulent channels, three-dimensional cylinders and turbulent separation bubbles. In all the cases, the discovered DRL-based strategies significantly outperform classical flow-control approaches. We conclude that DRL has tremendous potential for drag reduction in a wide range of complex turbulent-flow configurations.

• Dr. Zhengzhong Tu, Texas A&M University, USA, April 10, 2025, (12 - 1 pm ET). 

Title: Advancing Cooperative Perception Systems in Real-World Deployment: Challenges, Solutions, and Frontiers 

Abstract:  In this presentation, I will explore state-of-the-art advancements and practical solutions in vehicle-to-everything (V2X) cooperative perception for intelligent transportation systems. I begin by highlighting the limitationsof single-agent perception and show how connectivity can address critical challenges faced by automated vehicles inreal-world settings. Despite its potential, deploying cooperative perception systems introduces significant hurdles.This talk introduces several transformative contributions, including V2X-ViT (ECCV’22), a unified transformer architecture for robust perception in noisy environments, as well as CoBEVT (CoRL’22), an efficient vision transformer architecture tailored for BEV semantic segmentation using cost-effective camera-only strategies. We also introduceour latest work on using emerging vision architectures, Mamba, for real-time onboard detection. Furthermore, I willintroduce our recent framework accepted to ICLR’25 on designing a scalable and task-agnostic collaborative perception protocol, to facilitate heterogenous and secure mobility systems for the future networks. Lastly, I will talk about our latest efforts on benchmarking the trustworthiness of large vision-language models (VLMs) for autonomous driving, paving the way for future safe, robust, and private foundation models for autonomy. These advancements collectively push the boundaries of cooperative perception, offering scalable, efficient, and safety-critical solutions for autonomous systems in complex real-world environments.

• Dr. Deep Ray, University of Maryland, USA, April 17, 2025, (12 - 1 pm ET).

Title: An optimal Petrov-Galerkin framework for operator learning

Abstract: The optimal Petrov-Galerkin formulation to solve PDEs was first introduced in the 1980’s. Its goal was, given a trial basis, to approximate the finite-dimensional solution that is optimal with respect to a suitable norm. The theory is elegant and allows for the recovery of optimal convergence rates, even for problems where standard Galerkin methods fail. However, recovery of the optimal solution is contingent on being able to construct the optimal weighting functions associated with the trial basis. While explicit constructions are available for simple 1D and 2D problems, such constructions for a general multidimensional problem remain elusive. As a result, interest in this approach waned with the rise of alternative finite element strategies. In this talk, we revisit optimal weighting functions through the lens of deep learning. We propose an operator network framework called PG-VarMiON, that emulates the optimal Petrov-Galerkin weak form of the underlying PDE. Given a suitable trial basis and a norm, the PG-VarMiON approximates the optimal finite-dimensional while implicitly learning the optimal weighting functions. We derive an approximation error estimate for PG-VarMiON, highlighting the contributions of various error sources. Several numerical results for the advection-diffusion equation are presented to demonstrate the efficacy of the proposed method. By embedding the Petrov-Galerkin structure into the network architecture, PG- VarMiON exhibits greater robustness and improved generalization compared to other deep operator frameworks, particularly when training data is limited. The proposed approach harnesses the knowledge of traditional numerical methods to solve PDEs, paving the way for constructing mathematical-sound and efficient deep surrogates.

• Prof. Benjamin Peherstorfer, Courant Institute of Mathematical Sciences, New York University, April 24, 2025, (12 - 1 pm ET).

Title: DICE: Discrete inverse continuity equation for marginal trajectory matching

Abstract: The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the continuity equation, we use a variational problem to infer parameter- and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that a judicious discretization in time is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.

 Summer 2025 Break