Yasamin Jalalian

I am a PhD candidate in Applied and Computational Mathematics at Caltech, fortunate to be advised by Professor Houman Owhadi and co-mentored by Professor Franca Hoffmann.

I use tools from functional analysis and applied probability to study the theoretical foundations of data-driven methods for complex systems. In particular, I am interested in two broad categories:

Learning Differential Equations from Data

Theoretical and computational foundations for learning deterministic and stochastic governing equations, and their associated solution operators, in different data regimes

Function and Operator Approximation Theory

Error analysis for function and operator learning across different computational frameworks, including kernel methods and Gaussian processes

News

July 2026: I gave a talk at SIAM Annual Meeting (AN26) on "Learning Differential Equations from Scarce Data: A Kernel-Based Approach".
April 2026: I gave a talk at Southern California Applied Mathematics Symposium (SOCAMS) on "Learning Differential Equations from Scarce Data: A Kernel-Based Approach".
September 2025: Our preprint on "Data-efficient Kernel Methods for Learning Hamiltonian Systems" is on arXiv.
May 2025: I gave a talk at Machine Learning and Dynamical Systems Seminar on "Data-efficient Kernel Methods for Learning Equations: Part I - Learning PDEs" (video).
March 2025: Our preprint on "Data-efficient Kernel Methods for Learning Differential Equations and their Solution Operators: Algorithms and Error Analysis" is on arXiv.

Preprints

Data-efficient Kernel Methods for Learning Differential Equations and their Solution Operators: Algorithms and Error Analysis

Yasamin Jalalian, Juan Felipe Osorio Ramirez, Alexander Hsu, Bamdad Hosseini, Houman Owhadi

Abstract

In this work, we introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational cost, in terms of training procedures.

Our approach is backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation.
Numerical benchmarks demonstrate significant improvements in computational complexity and robustness while achieving one to two orders of magnitude improvements in terms of accuracy compared to state-of-the-art algorithms.
In comparison to equivalent neural net methods, our approach is significantly more robust to the choice of hyperparameters and does not require close human supervision during training.

arXiv

2025

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Yasamin Jalalian, Mostafa Samir, Boumediene Hamzi, Peyman Tavallali, Houman Owhadi

Abstract

In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from data. We present two approaches: a two-step method that reconstructs trajectories before learning the Hamiltonian, and a one-step method that jointly infers both.

Across several benchmark systems, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms two-step kernel-based baselines, particularly in scarce-data regimes, while preserving the conservation properties of Hamiltonian dynamics.
We also prove theoretical a priori error estimates, ensuring reliability of the learned models.