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Physics-Informed Neural Networks (PINNs) have gained popularity in solving nonlinear partial differential equations (PDEs) via integrating physical laws into the training of neural networks, making them superior in many scientific and…

We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…

Distance metric learning aims to learn from the given training data a valid distance metric, with which the similarity between data samples can be more effectively evaluated for classification. Metric learning is often formulated as a…

Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on…

Differentiable programming allows for derivatives of functions implemented via computer code to be calculated automatically. These derivatives are calculated using automatic differentiation (AD). This thesis explores two applications of…

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…

Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data$-$driven approaches…

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such…

Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…

Model-based paradigms for decision-making and control are becoming ubiquitous in robotics. They rely on the ability to efficiently learn a model of the system from data. Structured Mechanical Models (SMMs) are a data-efficient black-box…

Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical…

Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at…

Scientific machine learning (SciML) represents a significant advancement in integrating machine learning (ML) with scientific methodologies. At the forefront of this development are Physics-Informed Neural Networks (PINNs), which offer a…

Scaling laws have emerged as a unifying lens for understanding and guiding the training of large language models (LLMs). However, existing studies predominantly focus on the final-step loss, leaving open whether the entire loss dynamics…

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…

Recent advancements in Machine Learning (ML) have substantially improved its predictive and computational abilities, offering promising opportunities for surrogate modeling in scientific applications. By accurately approximating complex…

A novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure…

Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges, primarily due to the curse of dimensionality and the inherent limitations of traditional numerical…

The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and…