Course Description: This course focuses on modern scientific computing related to AI & PDE, with an emphasis on multi-task, general purpose, reusable neural networks for PDE problems. Rather than treating solvers as a collection of isolated numerical tasks, the course will examine emerging techniques for pre-training, multi-operator predictions, transferability, and generalization in which architectures learn representations across families of physical systems. Scientific foundation models will also be discussed, centering on capabilities of in-context adaptation, causal modeling, and model discovery across diverse scientific systems. We will cover the following:
Review of Operator Learning with an overview of Multi-Operator Learning
PROSE, PROSE-PDE: Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation
BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamics
ICON, VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction
PDEformer: Towards a Foundation Model for One-Dimensional Partial Differential Equations
MPP: Multiple Physics Pretraining for Physical Surrogate Models
DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training
Poseidon: Efficient Foundation Models for PDEs
Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers
Modeling aspects include: tokenization of equations, operators, and observations with in-context adaptation, causal reasoning, and generalizable surrogates.
Evaluation: Your grade is project based. The project will include a written report and a presentation.
This course is open to advanced undergraduates and all graduate students.
University of California, Los Angeles
Fall 2025: Math 266A: Applied Ordinary Differential Equations (Graduate)
Winter 2025: Math 269B: Advanced Numerical Analysis (Graduate)
Fall 2024: Math 277: Foundations of Machine Learning and Artificial Intelligence (Graduate)
Winter 2024: Math 285J Topics in Applied Math (Graduate)
Fall 2023: Math 266A: Applied Ordinary Differential Equations (Graduate)
Spring 2023: Math 151A: Applied Numerical Methods
Carnegie Mellon University
Fall 2022: Math 671: Computational Linear Algebra (MS in Data Analytics)
Fall 2021: Math 671: Computational Linear Algebra (MS in Data Analytics)
Fall 2021: Math 369: Numerical Analysis
Spring 2021: Math 599: LASSO Methods and Sparsity (Undergrad. Topics)
Fall 2020: Math 369: Numerical Analysis
Spring 2020: Math 860: Advanced Topics in Numerical Analysis (Graduate)
Fall 2018: Math 369: Numerical Analysis
Spring 2018: Math 660: Numerical Analysis (Graduate)
Fall 2017: Math 369: Numerical Analysis
Spring 2017: Math 660: Numerical Analysis (Graduate)
Fall 2016: Math 369: Numerical Analysis
Spring 2016: Math 369: Numerical Analysis