93 research outputs found
Accelerated Solutions of Coupled Phase-Field Problems using Generative Adversarial Networks
Multiphysics problems such as multicomponent diffusion, phase transformations
in multiphase systems and alloy solidification involve numerical solution of a
coupled system of nonlinear partial differential equations (PDEs). Numerical
solutions of these PDEs using mesh-based methods require spatiotemporal
discretization of these equations. Hence, the numerical solutions are often
sensitive to discretization parameters and may have inaccuracies (resulting
from grid-based approximations). Moreover, choice of finer mesh for higher
accuracy make these methods computationally expensive. Neural network-based PDE
solvers are emerging as robust alternatives to conventional numerical methods
because these use machine learnable structures that are grid-independent, fast
and accurate. However, neural network based solvers require large amount of
training data, thus affecting their generalizabilty and scalability. These
concerns become more acute for coupled systems of time-dependent PDEs. To
address these issues, we develop a new neural network based framework that uses
encoder-decoder based conditional Generative Adversarial Networks with ConvLSTM
layers to solve a system of Cahn-Hilliard equations. These equations govern
microstructural evolution of a ternary alloy undergoing spinodal decomposition
when quenched inside a three-phase miscibility gap. We show that the trained
models are mesh and scale-independent, thereby warranting application as
effective neural operators.Comment: 18 pages, 21 figures (including subfigures). Will be submitted to the
journal: "Computational Materials Science" soo
Physics-Informed Machine Learning for Data Anomaly Detection, Classification, Localization, and Mitigation: A Review, Challenges, and Path Forward
Advancements in digital automation for smart grids have led to the
installation of measurement devices like phasor measurement units (PMUs),
micro-PMUs (-PMUs), and smart meters. However, a large amount of data
collected by these devices brings several challenges as control room operators
need to use this data with models to make confident decisions for reliable and
resilient operation of the cyber-power systems. Machine-learning (ML) based
tools can provide a reliable interpretation of the deluge of data obtained from
the field. For the decision-makers to ensure reliable network operation under
all operating conditions, these tools need to identify solutions that are
feasible and satisfy the system constraints, while being efficient,
trustworthy, and interpretable. This resulted in the increasing popularity of
physics-informed machine learning (PIML) approaches, as these methods overcome
challenges that model-based or data-driven ML methods face in silos. This work
aims at the following: a) review existing strategies and techniques for
incorporating underlying physical principles of the power grid into different
types of ML approaches (supervised/semi-supervised learning, unsupervised
learning, and reinforcement learning (RL)); b) explore the existing works on
PIML methods for anomaly detection, classification, localization, and
mitigation in power transmission and distribution systems, c) discuss
improvements in existing methods through consideration of potential challenges
while also addressing the limitations to make them suitable for real-world
applications
A mean-field games laboratory for generative modeling
In this paper, we demonstrate the versatility of mean-field games (MFGs) as a
mathematical framework for explaining, enhancing, and designing generative
models. There is a pervasive sense in the generative modeling community that
the various flow and diffusion-based generative models have some common
foundational structure and interrelationships. We establish connections between
MFGs and major classes of flow and diffusion-based generative models including
continuous-time normalizing flows, score-based models, and Wasserstein gradient
flows. We derive these three classes of generative models through different
choices of particle dynamics and cost functions. Furthermore, we study the
mathematical structure and properties of each generative model by studying
their associated MFG's optimality condition, which is a set of coupled
forward-backward nonlinear partial differential equations (PDEs). The theory of
MFGs, therefore, enables the study of generative models through the theory of
nonlinear PDEs. Through this perspective, we investigate the well-posedness and
structure of normalizing flows, unravel the mathematical structure of
score-based generative modeling, and derive a mean-field game formulation of
the Wasserstein gradient flow. From an algorithmic perspective, the optimality
conditions of MFGs also allow us to introduce HJB regularizers for enhanced
training of a broad class of generative models. In particular, we propose and
demonstrate an Hamilton-Jacobi-Bellman regularized SGM with improved
performance over standard SGMs. We present this framework as an MFG laboratory
which serves as a platform for revealing new avenues of experimentation and
invention of generative models. This laboratory will give rise to a multitude
of well-posed generative modeling formulations and will provide a consistent
theoretical framework upon which numerical and algorithmic tools may be
developed.Comment: 38 pages, 10 figures. Version 4 includes derivation of the score
probability flo
Deep Learning And Uncertainty Quantification: Methodologies And Applications
Uncertainty quantification is a recent emerging interdisciplinary area that leverages the power of statistical methods, machine learning models, numerical methods and data-driven approach to provide reliable inference for quantities of interest in natural science and engineering problems. In practice, the sources of uncertainty come from different aspects such as: aleatoric uncertainty where the uncertainty comes from the observations or is due to the stochastic nature of the problem; epistemic uncertainty where the uncertainty comes from inaccurate mathematical models, computational methods or model parametrization. Cope with the above different types of uncertainty, a successful and scalable model for uncertainty quantification requires prior knowledge in the problem, careful design of mathematical models, cautious selection of computational tools, etc. The fast growth in deep learning, probabilistic methods and the large volume of data available across different research areas enable researchers to take advantage of these recent advances to propose novel methodologies to solve scientific problems where uncertainty quantification plays important roles. The objective of this dissertation is to address the existing gaps and propose new methodologies for uncertainty quantification with deep learning methods and demonstrate their power in engineering applications.
On the methodology side, we first present a generative adversarial framework to model aleatoric uncertainty in stochastic systems. Secondly, we leverage the proposed generative model with recent advances in physics-informed deep learning to learn the uncertainty propagation in solutions of partial differential equations. Thirdly, we introduce a simple and effective approach for posterior uncertainty quantification for learning nonlinear operators. Fourthly, we consider inverse problems of physical systems on identifying unknown forms and parameters in dynamical systems via observed noisy data.
On the application side, we first propose an importance sampling approach for sequential decision making. Second, we propose a physics-informed neural network method to quantify the epistemic uncertainty in cardiac activation mapping modeling and conduct active learning. Third, we present an anto-encoder based framework for data augmentation and generation for data that is expensive to obtain such as single-cell RNA sequencing
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
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