Host: Zecheng Zhang.
Room: Love 0231 or online.
Zoom Link: will be avaiable up on request.
Time: 03:05 pm to 04:05 pm. It is weekly on Tuesday, see below for the detailed schedule.
| Date | Speaker | Abstract of the Talk |
|---|---|---|
| Aug 27 | Zecheng Zhang | Introduction to the ACM seminar of Fall 2024, speaker informations, and what to know. |
| Sept 3 | Jingmin Sun (CMU) | In this talk, we develop and explore a foundational model for solving Partial Differential Equations (PDEs) with an emphasis on building a versatile, robust framework capable of addressing multiple PDE operators simultaneously. Our objective is to create a single model that not only handles various operators but also demonstrates the ability to generalize to new, unseen physical phenomena in a "zero-shot" manner. We present numerical examples to demonstrate the model's ability to generalize physical phenomena in a "zero-shot" manner. We introduce LeMON, a Learning to Learn Multi-Operator Network pipeline, which integrates both pre-training and fine-tuning processes. Furthermore, we investigate a new scaling law specifically tailored to PDE foundation models, providing insights into optimizing model performance as it scales. Finally, we improve the LeMON pipeline by integrating LoRA and meta-learning strategies. |
| Sept 10 | Bryce Morsky (FSU) | The replicator equation is widely used in modelling biological, economic, and social systems. It traditionally assumes proportional selection and that replicators earn mean payoffs. In this talk, I will present extensions that feature payoff distributions and truncation selection, where only replicators with fitness above a threshold survive and reproduce. We can distinguish between two types of truncation selection: replicators below a fixed fitness threshold are culled, or a bottom proportion of the population is culled. I will present analyses of these equations, comparing them to the standard replicator equation. |
| Sept 17 | Wenqi Cui (Caltech and NYU) | This talk will describe how to bridge the gap between learning and safety-critical constraints through structured neural networks guided by control theory and the physics of energy systems. Using Lyapunov theory, I will show how we can extract stabilizing controller structures for transient stability problems, and show how to parameterize the structures by neural networks. Then I will further show how we can achieve provable guarantees on steady-state optimal resource allocation and adapt to time-varying loads and renewables. The extension of the framework to broader networked systems will also be discussed. |
| Sept 24 | Rishi Sonthalia (Boston College) | A fundamental problem in machine learning is understanding the effect of early stopping and mini-batching on the parameters obtained and the generalization capabilities of the trained model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. We analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with a learning rate schedule $\eta_k$ and a finite time horizon $T$, the early stopped solution $\beta_T$ is equivalent to the minimum norm solution for a generalized ridge regularized problem. We also prove that early stopping is beneficial for generic data with arbitrary covariance spectrum and various learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate. We also study the discrete dynamics of mini-batch gradient descent with random reshuffling for least squares regression. We show that the error dynamics and generalization error depends on a sample cross-covariance matrix $\mathbf{Z}$ between the original features $\mathbf{X}$ and a set of new features $\widetilde{\mathbf{X}}$, in which each feature is modified by the mini-batches that appear before it during the learning process in an averaged way. |
| Oct 1 | Haoyang Qian (FSU) | Polarization significantly influences societal divisions across economic, political, religious, and ideological lines. Understanding these mechanisms is key to devising strategies to mitigate such divisions and promote depolarization. Our study examines how asymmetric opinion perception, modeled through nonlinear incidence terms, affects polarization and depolarization within structured communities. We demonstrate that such asymmetry leads to explosive polarization and causes a hysteresis effect responsible for abrupt depolarization. We develop a mean-field approximation to explain how nonlinear incidence results in first-order phase transitions and the nature of bifurcations. This approach also helps us understand how opinions polarize according to underlying social network communities. Numerical simulations corroborate the analytical findings. |
| Oct 8 | Ali Kara (FSU) | Mean-Field Control: Decentralization and Learning. |
| Oct 15 | TBA | TBA |
| Oct 29 | Hannah Lu (UT Austin) | Data-driven modeling of complex systems is a rapidly evolving field facilitated by the concurrent rise of data science. To alleviate the prohibitively expensive computational costs of repeated full-model simulations in uncertainty quantification, data-driven modeling is often used to describe the behaviors of the complex system by predicting the quantities of interest directly. In this talk, I will present my contributions to this field with an emphasis on (1) improving model performance by using physics-aware machine learning techniques, (2) quantifying uncertainties in the system’s response, and (3) inferring the key parameters of the physics-based models from measured data. Examples of applications will be focused on large-scale geological carbon sequestration—an important strategy for reducing greenhouse gas emissions to the atmosphere and mitigating climate change. The objective is to develop a convenient computing toolbox to provide more accurate scientific information at cheaper computational costs for better environmental management and decision-making. |
| Nov 5 | Fengjiao Liu (FSU) | In recent years, the need to quantify and control the uncertainty in physical systems has prompted a burgeoning interest in studying the evolution of the distribution of the trajectories of stochastic systems. A special case of this point of view is covariance steering, which aims to control the covariance of the state of a stochastic system from a given initial covariance to a desired terminal covariance on a finite time horizon. In this talk, we first study the controllability of the state covariance of a linear stochastic system. Then, we give an optimal control law for covariance steering with a quadratic cost and discuss how to efficiently compute the optimal control law via semi-definite programming. Lastly, we provide an interesting byproduct on Riccati differential/difference equations, which are widely used in optimal control. |
| Nov 12 | Sui Tang (UCSB) | Interacting particle systems showcase a variety of collective behaviors and are fundamental to many scientific and engineering domains, such as the flocking of birds and the milling of fish. These systems are typically modeled using differential equations to elucidate how individual behaviors drive collective dynamics, an essential inquiry across multiple disciplines. Although recent theoretical and numerical studies have successfully replicated many qualitative collective patterns seen in nature, there is still a notable deficiency in quantitatively matching these models with empirical data. We explore the data-driven discovery of latent interaction kernels from observed trajectory data in particle and agent-based systems. We discuss recent findings in stochastic systems where interaction kernels are derived from pairwise distances and introduce a nonparametric inference strategy using a regularized maximum likelihood estimator for precise kernel estimation. We show this approach can achieve near-optimal convergence rates, regardless of the state space dimensionality when dealing with multiple trajectory data. Additionally, we conduct error analysis related to discrete-time observations and validate our methodology through numerical experiments on models such as stochastic opinion dynamics and the Lennard-Jones potential. Moreover, we also consider microscopic models and advance our approach to estimate nonlocal interaction potentials in aggregation-diffusion equations from noisy data, using sparsity-promoting techniques. This research is conducted in collaboration with Fei Lu, Mauro Maggioni, Jose A. Carrillo, Gissell Estrada-Rodriguez, and Laszlo Mikolas. |
| Nov 19 | Weiqi Chu (Umass) | Consider a complex system of a vast number of interacting agents, which can give rise to emergent collective behaviors such as bird flocking, fish swarming, and opinion formation — phenomena that cannot be obtained by studying individual agents alone. However, high-dimensional systems present significant challenges for inference tasks, particularly when data availability is limited. In this presentation, I will discuss how a mean-field approach can be used to derive mean-field models from agent-based dynamics and to infer interaction kernels with limited data observation. I will also address the PDE-constrained optimization method employed and highlight the identifiability challenges that commonly arise in such inference problems. |
| Dec 3 | TBA | TBA |