Yasamin Jalalian

I am a PhD candidate in Applied and Computational Mathematics at Caltech, where I am fortunate to be advised by Professors Houman Owhadi and Franca Hoffmann. Prior to joining Caltech, I earned my bachelor’s degree in Mathematics and Computer Science from École Polytechnique.

Research Interests

My research lies at the intersection of mathematical analysis and machine learning, with a primary focus on the theoretical foundations of data-driven methods for complex systems. I am particularly interested in kernel methods and Gaussian processes for learning and solving partial differential equations, with an emphasis on analyzing error bounds and function approximation properties across various model architectures and problem settings.

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

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 mathematically interpretable and 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

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. 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. Moreover, we prove theoretical a priori error estimates, ensuring reliability of the learned models.

arXiv

2025

Yasamin Jalalian

Research Interests

Preprints

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

Data-efficient Kernel Methods for Learning Hamiltonian Systems

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