arXiv subjects

Christian Moya

Publications and source records attributed to Christian Moya.

27 records · Page 2Linked to original sources

arXiv2023

Bayesian deep operator learning for homogenized to fine-scale maps for multiscale PDE

We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using a limited number (and possibly noisy) observations of the fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.

math.NA↗

arXiv2023

Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profiles

This paper designs surrogate models with uncertainty quantification capabilities to improve the thermal performance of rib-turbulated internal cooling channels effectively. To construct the surrogate, we use the deep operator network (DeepONet) framework, a novel class of neural networks designed to approximate mappings between infinite-dimensional spaces using relatively small datasets. The proposed DeepONet takes an arbitrary continuous rib geometry with control points as input and outputs continuous detailed information about the distribution of pressure and heat transfer around the profiled ribs. The datasets needed to train and test the proposed DeepONet framework were obtained by simulating a 2D rib-roughened internal cooling channel. To accomplish this, we continuously modified the input rib geometry by adjusting the control points according to a simple random distribution with constraints, rather than following a predefined path or sampling method. The studied channel has a hydraulic diameter, Dh, of 66.7 mm, and a length-to-hydraulic diameter ratio, L/Dh, of 10. The ratio of rib center height to hydraulic diameter (e/Dh), which was not changed during the rib profile update, was maintained at a constant value of 0.048. The ribs were placed in the channel with a pitch-to-height ratio (P/e) of 10. In addition, we provide the proposed surrogates with effective uncertainty quantification capabilities. This is achieved by converting the DeepONet framework into a Bayesian DeepONet (B-DeepONet). B-DeepONet samples from the posterior distribution of DeepONet parameters using the novel framework of stochastic gradient replica-exchange MCMC.

physics.flu-dyn↗

arXiv2022

DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-shot Transfer the Dynamic Response of Networked Systems

This paper develops a Deep Graph Operator Network (DeepGraphONet) framework that learns to approximate the dynamics of a complex system (e.g. the power grid or traffic) with an underlying sub-graph structure. We build our DeepGraphONet by fusing the ability of (i) Graph Neural Networks (GNN) to exploit spatially correlated graph information and (ii) Deep Operator Networks~(DeepONet) to approximate the solution operator of dynamical systems. The resulting DeepGraphONet can then predict the dynamics within a given short/medium-term time horizon by observing a finite history of the graph state information. Furthermore, we design our DeepGraphONet to be resolution-independent. That is, we do not require the finite history to be collected at the exact/same resolution. In addition, to disseminate the results from a trained DeepGraphONet, we design a zero-shot learning strategy that enables using it on a different sub-graph. Finally, empirical results on the (i) transient stability prediction problem of power grids and (ii) traffic flow forecasting problem of a vehicular system illustrate the effectiveness of the proposed DeepGraphONet.

cs.LG↗

arXiv2022

DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories

This paper proposes a new data-driven method for the reliable prediction of power system post-fault trajectories. The proposed method is based on the fundamentally new concept of Deep Operator Networks (DeepONets). Compared to traditional neural networks that learn to approximate functions, DeepONets are designed to approximate nonlinear operators. Under this operator framework, we design a DeepONet to (1) take as inputs the fault-on trajectories collected, for example, via simulation or phasor measurement units, and (2) provide as outputs the predicted post-fault trajectories. In addition, we endow our method with a much-needed ability to balance efficiency with reliable/trustworthy predictions via uncertainty quantification. To this end, we propose and compare two methods that enable quantifying the predictive uncertainty. First, we propose a \textit{Bayesian DeepONet} (B-DeepONet) that uses stochastic gradient Hamiltonian Monte-Carlo to sample from the posterior distribution of the DeepONet parameters. Then, we propose a \textit{Probabilistic DeepONet} (Prob-DeepONet) that uses a probabilistic training strategy to equip DeepONets with a form of automated uncertainty quantification, at virtually no extra computational cost. Finally, we validate the predictive power and uncertainty quantification capability of the proposed B-DeepONet and Prob-DeepONet using the IEEE 16-machine 68-bus system.

math.NA↗

arXiv2021

Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs

The Deep Operator Networks~(DeepONet) is a fundamentally different class of neural networks that we train to approximate nonlinear operators, including the solution operator of parametric partial differential equations (PDE). DeepONets have shown remarkable approximation and generalization capabilities even when trained with relatively small datasets. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs very often in practice. To enable DeepONets training with noisy data, we propose using the Bayesian framework of replica-exchange Langevin diffusion. Such a framework uses two particles, one for exploring and another for exploiting the loss function landscape of DeepONets. We show that the proposed framework's exploration and exploitation capabilities enable (1) improved training convergence for DeepONets in noisy scenarios and (2) attaching an uncertainty estimate for the predicted solutions of parametric PDEs. In addition, we show that replica-exchange Langeving Diffusion (remarkably) also improves the DeepONet's mean prediction accuracy in noisy scenarios compared with vanilla DeepONets trained with state-of-the-art gradient-based optimization algorithms (e.g. Adam). To reduce the potentially high computational cost of replica, in this work, we propose an accelerated training framework for replica-exchange Langevin diffusion that exploits the neural network architecture of DeepONets to reduce its computational cost up to 25% without compromising the proposed framework's performance. Finally, we illustrate the effectiveness of the proposed Bayesian framework using a series of experiments on four parametric PDE problems.

math.NA↗

arXiv2021

DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networks

Deep learning-based surrogate modeling is becoming a promising approach for learning and simulating dynamical systems. Deep-learning methods, however, find very challenging learning stiff dynamics. In this paper, we develop DAE-PINN, the first effective deep-learning framework for learning and simulating the solution trajectories of nonlinear differential-algebraic equations (DAE), which present a form of infinite stiffness and describe, for example, the dynamics of power networks. Our DAE-PINN bases its effectiveness on the synergy between implicit Runge-Kutta time-stepping schemes (designed specifically for solving DAEs) and physics-informed neural networks (PINN) (deep neural networks that we train to satisfy the dynamics of the underlying problem). Furthermore, our framework (i) enforces the neural network to satisfy the DAEs as (approximate) hard constraints using a penalty-based method and (ii) enables simulating DAEs for long-time horizons. We showcase the effectiveness and accuracy of DAE-PINN by learning and simulating the solution trajectories of a three-bus power network.

cs.LG↗

arXiv2018

Application of Correlation Indices on Intrusion Detection Systems: Protecting the Power Grid Against Coordinated Attacks

The future power grid will be characterized by the pervasive use of heterogeneous and non-proprietary information and communication technology, which exposes the power grid to a broad scope of cyber-attacks. In particular, Monitoring-Control Attacks (MCA) --i.e., attacks in which adversaries manipulate control decisions by fabricating measurement signals in the feedback loop-- are highly threatening. This is because, MCAs are (i) more likely to happen with greater attack surface and lower cost, (ii) difficult to detect by hiding in measurement signals, and (iii) capable of inflicting severe consequences by coordinating attack resources. To defend against MCAs, we have developed a semantic analysis framework for Intrusion Detection Systems (IDS) in power grids. The framework consists of two parts running in parallel: a Correlation Index Generator (CIG), which indexes correlated MCAs, and a Correlation Knowledge-Base~(CKB), which is updated aperiodically with attacks' Correlation Indices (CI). The framework has the advantage of detecting MCAs and estimating attack consequences with promising runtime and detection accuracy. To evaluate the performance of the framework, we computed its false alarm rates under different attack scenarios.

cs.CR↗

arXiv2018

Developing a Correlation Indices to Identify Coordinated Cyber-Attacks on Power Grids

Increasing reliance on Information and Communication Technology~(ICT) exposes the power grid to cyber-attacks. In particular, Coordinated Cyber-Attacks (CCAs) are considered highly threatening and difficult to defend against, because they (i) possess higher disruptiveness by integrating greater resources from multiple attack entities, and (ii) present heterogeneous traits in cyber-space and the physical grid by hitting multiple targets to achieve the attack goal. Thus, and as opposed to independent attacks, whose severity is limited by the power grid's redundancy, CCAs could inflict disastrous consequences, such as blackouts. In this paper, we propose a method to develop Correlation Indices to defend against CCAs on static control applications. These proposed indices relate the targets of CCAs with attack goals on the power grid. Compared to related works, the proposed indices present the benefits of deployment simplicity and are capable of detecting more sophisticated attacks, such as measurement attacks. We demonstrate our method using measurement attacks against Security Constrained Economic Dispatch.

eess.SY↗

arXiv2018

A Protection Method in Active Distribution Grids with High Penetration of Renewable Energy Sources

A protection method in active distribution networks is proposed in this paper. In active distribution systems, fault currents flow in multiple directions and presents a varying range of value, which poses a great challenge of maintaining coordination among protective devices on feeders. The proposed protection method addresses this challenge by simultaneously adjusting DG's output power and protection devices' settings in pre-fault networks. Comparing to previous protection solutions, the proposed method considers the influences from renewable DG's intermittency, and explores the economic and protection benefits of DG's active participation. The formulation of proposed method is decomposed into two optimization sub-problems, coupling through the constraint on fuse-recloser coordination. This decomposed mathematical structure effectively extinguishes the non-linearity arising from reclosers' time-current inverse characteristics, and greatly reduces computation efforts.

math.OC↗