Prof. Karen Willcox and I authored a piece for SIAM News about our perspectives and recent work on digital twins. Here is a short excerpt from the article:

"The early successes of digital twin deployment point to the idea’s value and potential impact. Now is the time for the applied mathematics and computational science communities to develop the rigorous mathematical underpinnings and scalable algorithms that will take digital twins to the next level. [For] inspiration we can look to the evolution of the finite element method from an expert-driven approach that required specialization for each different application to a broadly applicable analysis and design tool that is now in the hands of every engineer. This evolution has been enabled by foundational mathematical theory, computing scalability achieved through a combination of hardware and algorithmic advances, and flexible software implementations.

In order to advance digital twins to a similar level of maturity and accessibility, our community’s work in a variety of areas—including physics-based modeling, inverse problems, data assimilation, uncertainty quantification, optimal control, optimal experimental design, surrogate modeling, scientific machine learning, scalable algorithms, and scientific software—has an important role to play."

On May 6, 2021 I successfully defended my doctoral thesis entitled Mathematical and Computational Foundations to Enable Predictive Digital Twins at Scale. You can find a copy of my full PhD Thesis here, as well as a pdf copy of my thesis defense presentation slides here.

A big thank-you to my PhD advisor, Karen Willcox, and my thesis defense committee Prof. John-Paul Clarke, Prof. Youssef Marzouk, Dr. Kevin Carlberg, and Dr. David Knezevic!

Our latest paper, published in Nature Computational Science proposes a unified mathematical and computational foundation for digital twins. Below you can find a high-level summary of our motivation and approach for this work. For technical details check out our full paper (open-access preprint available here), or for a more complete picture of how this work fits into my broader digital twin research check out my PhD thesis.

This work was previously presented at the 2021 AIAA SciTech Forum, and the 2021 SIAM Conference on Computational Science and Engineering.  

Updates:
[January 31 2022] Our paper was selected for the Nature Computational Science one year anniversary collection!
[September 1 2021] This work was featured in a SIAM News article (I am a co-author)
[July 20 2021] This work was featured by Tech Briefs
[July 16 2021] Our MIT News article was featured by the Department of Energy Office of Science
[June 21 2021] This work was featured on Wevolver's The Next Byte Podcast.
[June 15 2021] This work was featured by MIT News! Check out the full story and my interview here.
[May 20 2021] This work was featured by UT Austin News!
[May 20 2021] Our paper was the Nature Computational Science Editor Pick!

Michael Kapteyn and colleagues propose a formal mathematical foundation based on probabilistic graphical models for digital twins, supporting principled data assimilation, optimal control, and end-to-end uncertainty quantification. Check it out: https://t.co/8lzY1v1gde pic.twitter.com/frEAp61msx

— Nature Computational Science (@NatComputSci) May 20, 2021

A Mathematical and Computational Foundation for Predictive Digital Twins

Current state-of-the-art digital twins are largely the result of custom implementations that require considerable deployment resources and a high level of expertise. In this work we propose a unifying mathematical foundation for digital twins, with the goal of moving from the one-off digital twin to accessible and robust digital twin implementations at scale. Specifically, we define the combined asset-twin system in terms of six key quantities, which are shown in the figure below.

Defining the system in this way enables us to model the evolution of the physical asset and its digital twin over time. The digital twin learns about the state of the physical asset by assimilating incoming observational data. This information is used to update the internal digital twin models, which are then evaluated to provide accurate analysis and predictions. This enables the digital twin to positively influence the physical asset by informing and/or issuing intelligent control inputs. We model this two-way coupled system using the probabilistic graphical model below, which illustrates the dependencies between quantities in the system as it evolves over time.

Graphical models are a powerful tool for probabilistic representation, inference, and learning, and have been successfully applied in applications ranging from robotics and computer vision, to speech recognition and medical diagnosis. Our proposed probabilistic graphical model serves as a foundation for defining and modeling a digital twin, as well as deriving computationally efficient algorithms for model updating, prediction, and optimal control. More information about the proposed probabilistic graphical model formulation, as well as a demonstration of how we have used the model to perform principled and scalable experimental calibration of a UAV digital twin, can be found in our research paper.

This week (November 7 - 13, 2020) I attended the 2020 INFORMS annual meeting. I was pleased to be invited to present a talk entitled "From Physics-based Models to Predictive Digital Twins via Interpretable Machine Learning" in the Dynamic Data-driven Application Systems (DDDAS) mini-track.

In this talk, I discussed our research on combining physics-based models and interpretable machine learning techniques in order to automatically discover and implement goal-oriented sensing strategies. Details of my presentation, as well as a link to the presentation slides, is given below.

Slides:

This week (October 2-4, 2020), I attended the third international conference on Dynamic Data-driven Application Systems (DDDAS). Recorded presentations and abstracts can be found on the conference website, while papers can be found in the conference proceedings, published in Springer's Lecture Notes in Computer Science book series.

I was a co-author on two papers related to my research developing a UAV digital twin:

Predictive digital twins: Where dynamic data-driven learning meets physics-based modeling
[Presentation recording] [Slides] [Paper]
My advisor Professor Karen Willcox presented an opening keynote talk that broadly describes our philosophy and research on digital twins.

A Hardware Testbed for Dynamic Data-Driven Aerospace Digital Twins
[Presentation recording] [Slides] [Paper]
This paper presents the UAV hardware platform that our group has been developing in order to test and demonstrate our digital twin and self-aware UAV models and algorithms.

My PhD work was featured in an article at the Oden Institute for Computational Engineering and Sciences. The article is beautifully written and provides a great introduction to my research!

"In the not too distant future, we can expect to see our skies filled with unmanned aerial vehicles (UAVs)...In such a world, there will also be a digital twin for each UAV in the fleet: a virtual model that will follow the UAV through its existence, evolving with time."

This post introduces my PhD work developing a predictive digital twin of an unmanned aerial vehicle (UAV). I presented this work at the AIAA Scitech conference in January 2020, where our paper was awarded the Southwest Research Institute Student Paper Award in Non-Deterministic Approaches. The paper was also awarded the AIAA Multi-Disciplinary Optimization (MDO) Best Paper award for 2020.

For more details, see the full paper "Toward predictive digital twins via component-based reduced-order models and interpretable machine learning", or the slides for my presentation at AIAA Scitech. A preprint that includes more detail on the interpretable machine learning side of this work is also available on arXiv. More information can also be found on our research group website.

Motivation

Unmanned aerial vehicles (UAVs) are poised to become ubiquitous in a wide range of industries, with applications including inspection, agriculture, disaster relief, package delivery, and urban air mobility (a.k.a. air taxis!). It is critical that these UAVs are capable of monitoring their structural health. A self-aware UAV is capable of detecting, characterizing and responding intelligently to changes in it's structural health. This self-awareness will increase the safety and reliability of UAVs by avoiding structural failure, while also improving the efficiency and utilization of the UAV.

We seek to enable this capability by creating a Digital Twin of the UAV: a comprehensive virtual model that evolves alongside the UAV and captures it's structural health throughout its existence.

Project Overview

I am working on developing computational methods and algorithms that enable creation of a Predictive Digital Twin. We combine scientific machine learning with predictive physics-based models. Component-based reduced-order modeling makes the approach computationally efficient and scalable, while interpretable machine learning methods enable reliable data-driven decision-making. We combine these technologies to build a structural Digital Twin that is updated in near real-time based on sensed structural data, and is used to drive dynamic flight planning decisions.

Hardware Platform

Although the methods we develop can be applied to a wide range of physical assets, our testbed for this research is a custom-built 12ft wingspan fixed-wing UAV, developed in collaboration with Aurora Flight Sciences. This UAV is outfitted with a suite of structural sensors such as strain gauges, accelerometers, and high frequency vibration sensors.

Physics-based digital twin

At the heart of our digital twin is a library of physics-based models, each representing a different structural state. Through a collaboration with Akselos, we adopt a component-based reduced-order modeling approach, so that these physics-based models are accurate and fast to evaluate, even at the scale of the full UAV structure.
We model different damage states by creating multiple versions of each component in the model. Each version has a different damage state. In this example we create five copies of two components in the right wing of the UAV. Each copy has a reduction in stiffness of between 0% (pristine case) and 80% (worst damage case).
In flight, we use on-board structural sensor data to estimate which model best matches the current state of the UAV, and use this model in the digital twin. This ensures that the digital twin is constantly updated to reflect the current state of the UAV.

Interpretable Machine Learning

We use interpretable machine learning to train an optimal classification tree that predicts which model from the library best matches a set of structural measurements. These trees partition the space of sensor measurements so that each resulting region corresponds to a particular damage state.
When a new sensor measurement is acquired, we use the classification tree to decide which damage state best matches the data. The classification tree is interpretable because explicitly characterizes decision boundaries and it naturally enables sparse sensing.

Dynamic Decision Making

We demonstrate the benefits of our approach on an illustrative UAV scenario. In this scenario, the UAV must fly safely through a set of obstacles to a goal location while accumulating structural degradation. The UAV must choose either an aggressive flight path or a more conservative path around each obstacle. The aggressive path is faster, but requires the UAV to make sharp turns that subject the UAV to high structural loads. In contrast, the more conservative route is slower but subjects the UAV to lower structural loads. In pristine condition, the aircraft structure can safely withstand the higher loading, but as the aircraft wing accumulates damage or degradation the high load may lead to structural failure.

Our self-aware UAV uses the rapidly updating digital twin in order to monitor its evolving structural state and dynamically estimate its flight capability. Based on these capability estimates the UAV is able to dynamically replan the mission in order to maximize speed while avoiding structural failure.

Thanks for reading! If this article was useful for your own work, please be sure to cite our research paper:

Kapteyn, M., Knezevic, D. and Willcox, K., Toward predictive digital twins via component-based reduced-order models and interpretable machine learning. In proceedings of 2020 AIAA Scitech Forum and Exhibition, Orlando, FL, January 2020.

Prof. Karen Willcox (my PhD advisor) gave an invited talk at SC19, entitled "Predictive Data Science for Physical Systems: From Model Reduction to Scientific Machine Learning". This talk offers a great high-level overview of the research interests of our group, and how we view the interplay between data and physics-based models. The talk also highlights some of my PhD work enabling a Digital Twin for a self-aware unmanned aerial vehicle. The presentation slides for this talk are available on Professor Willcox's website. Here's a teaser:

This week (July 28 - August 1, 2019) I attended the 15th US National Congress on Computational Mechanics (USNCCM) in Austin, TX. I was pleased to be invited to present a talk entitled "Toward a Self-Aware Aircraft: Data-Driven Decisions, Adaptive Reduced Models, and Digital Twins" in the minisymposium on "Model Order Reduction for Computational Continuum Mechanics". Details of my presentation, as well as a link to the presentation slides, is given below. Thank you to the conference organizers, and to the minisymposium chairs for the opportunity to present, and for hosting a great conference.

Slides:

Abstract:

This talk presents a computational framework that combines reduced-order models (ROMs) with online sensor information in order to enable the next generation of self-aware unmanned aerial vehicles (UAVs). A self-aware UAV is one that maintains knowledge of its own internal state and acts accordingly[1]. The internal state we consider is the vehicle’s structural health. The vehicle uses sensor data to detect changes in its structural health, and responds intelligently by adapting the way it performs. This allows the vehicle to fly aggressively when it is healthy, while avoiding structural failures by becoming more conservative as it degrades. Such capability has the potential to improve performance over the full vehicle lifecycle, while also reducing structural health monitoring and maintenance costs.

We create a Digital Twin of the UAV, using a combination of physics-based and empirical models that represent the key aspects of the vehicle’s performance. A core part of this Digital Twin is a component-based ROM of the aircraft structure[2]. This structural model is evolved and adapted over the lifecycle of the vehicle, such that it reflects the vehicle’s current structural health after each mission. The component-based ROM paradigm provides flexibility in structural defect modeling, while also permitting efficient recalibration in the face of dynamic data. The Digital Twin is used to construct a library of ROMs that represent possible future structural states. Each entry in the library is mapped offline to a corresponding estimate of vehicle capability. This library is used in flight, as online sensor data is assimilated, to rapidly classify the evolving structural state over time. The near real-time estimation of the vehicle’s structural state enables dynamic mission re-planning in response to in-flight events such as damage to the UAV’s wing[3].

This talk will focus on computational aspects of this project, which also includes a 12-foot wingspan flight-test vehicle being constructed by the Willcox Research Group and Aurora Flight Sciences.

[1] Allaire, D., G. Biros, J. Chambers, O. Ghattas, D. Kordonowy, and K. Willcox. “Dynamic Data Driven Methods for Self-Aware Aerospace Vehicles.” Procedia Computer Science 9 (2012): 1206–1210.
[2] Ballani, J., Huynh, D. B. P., Knezevic, D. J., Nguyen, L., & Patera, A. T. (2018). A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities. Computer Methods in Applied Mechanics and Engineering, 329, 498-531.
[3] Singh, V., & Willcox, K. E. (2017). Methodology for Path Planning with Dynamic Data-Driven Flight Capability Estimation. AIAA Journal, 2727-2738.

This post showcases a class project I completed for MIT course 16.S398: Visual Navigation for Autonomous Vehicles (VNAV) in Fall 2018, taught by Prof. Luca Carlone. This was a really fun project, and provided great hands-on experience with trajectory optimization, controller design, simulation, and hardware testing for quadrotors!

For this project we sought to improve on the native waypoint tracking controller provided within the PX4 autopilot stack. To do this, we combined a minimum-snap trajectory optimization algorithm with a geometric controller designed for complex quadrotor maneuvers.

The code for this project was written in C++ using ROS, and built on the mavros package. Our extensions to the mavros_controllers package are available on github. Simulation studies were performed using Gazebo, while final testing was performed using an Intel Aero drone using the MIT high bay testing area.

Below is a video abstract of the project. For more details check out our summary slides on google slides.

Cover image credit: https://docs.px4.io/master/en/complete_vehicles/intel_aero.html

In 2014/15 I undertook a 3 month research internship at the Okinawa Institute of Science and Technology (OIST). During this time I worked under Professor Tsumoru Shintake on the early stages of what is now known as the OIST Wave Energy Project.

During this time we designed, manufactured, and tested a novel design for a wave energy converter: An adaptive-pitch wave energy turbine. The blades of this turbine were designed to be flexible, so that they could adapt their pitch depending on wave strength in order to maximize efficiency and protect against strong storm conditions.

Following my time at OIST, development of the wave energy turbine has continued rapidly. The group now has wave-energy turbines installed on the coast of the Maldives and on the coast of Okinawa!