The fast development of machine learning (ML) has resulted in explosive advancement in many aspects of science and engineering. In particular, deep learning methods allow one to understand highly complex systems that are almost impossible to study by the traditional methods. Inevitably, uncertainty and noise are ubiquitous in such complex systems. And the design and application of ML methods need to take into account of these uncertainties. On the other hand, modern ML methods also provide new perspectives and algorithm design possibilities for uncertainty quantification. It is the interface of ML and UQ that this mini-symposium focuses on. Our aim is to bring together a group of experts in ML and UQ and to foster knowledge exchange. Our primary focus is on two front: (1) how uncertainty can be quantified in modern ML methods; and (2) how modern ML methodology can help UQ to further our understanding and advance algorithm design.
10:30
Uncertainty quantification in chaotic systems
Tiernan Casey | Sandia National Laboratories | United States
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Authors:
Tiernan Casey | Sandia National Laboratories | United States
Habib Najm | Sandia National Laboratories | United States
Model parameter estimation using inference from data relies on the robust definition of a distance metric between data sets, typically chosen to be some vector norm computed using a logical ordering of the elements. For chaotic dynamical systems, residuals defined using point-wise distances between temporally collocated data points generated from numerical simulations and ground truth are no longer useful as small numerical perturbations result in disparate trajectories, even when the underlying parameters are identical. Rather than compare the data directly, we explore ideas to compare the overall trajectory structure as a means to define distances, and to make this effort tractable by operating with coordinates defined by the low-dimensional manifold where the trajectory data lives, estimated using unsupervised machine learning approaches. These methods are compared to approaches using concepts from state estimation with filtering.
11:00
Governing Equations Recovery for Systems with Uncertainties
Zhen Chen | Ohio State University | United States
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Authors:
Zhen Chen | Ohio State University | United States
John Jakeman | Sandia National Laboratories | United States
Tong Qin | Ohio State University | United States
Kailiang Wu | Ohio State University | United States
We present a numerical framework for recovering unknown governing differential equations subject to uncertain parameters. The framework utilizes deep neural networks, which incorporate the uncertain parameters in the systems as part of the modeling process. The learned system is then able to predict the system behavior and provide UQ analysis. Upon discuss the detail of the network structure, we then present an extensive set of numerical examples demonstrate the properties and effectiveness of our equation recovery algorithms.
11:30
Variational system identification of partial differential equations governing pattern formation: Inference with sparse, noisy and incomplete data
Krishna Garikipati | University of Michigan | United States
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Xun Huan | University of Michigan | United States
Krishna Garikipati | University of Michigan | United States
Zhenlin Wang | University of Michigan | United States
We address the problem of system identification of partial differential equations (PDEs) by building on our recent work, in which we demonstrated the ability to distinguish between competing mathematical models of pattern forming physics. The motivation comes from developmental biology, where pattern formation is central to the development of any multicellular organism, and from materials physics, where phase transitions similarly lead to microstructure. In both these fields there is a collection of nonlinear, parabolic PDEs that, over suitable parameter intervals and regimes of physics, can resolve the patterns or microstructures with comparable fidelity. This observation frames the question of which PDE best describes the data at hand. This question is particularly compelling because the identification of the governing PDE, with quantifiable uncertainty, immediately delivers insights to the physics underlying the systems. While building on recent work that uses stepwise regression, we present advances that leverage the variational framework and statistical tests. In previous work [1] we studied the influences of variable fidelity and noise in the data. Building upon this foundation, in this talk we present our latest developments for robust system identification with sparse and incomplete data using the variational basis.
12:00
Kernel mode decomposition and regression networks
Houman Owhadi | California Institute of Technology | United States
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Houman Owhadi | California Institute of Technology | United States
We introduce programmable and interpretable regression networks for pattern recognition. The programming of these networks is achieved by assembling elementary modules decomposing and recomposing kernels and data across levels of abstraction. We illustrate the proposed framework by programming regression networks solving (to near machine precision and with convergence guarantees) empirical mode decomposition problems with noisy signals, unknown non-trigonometric periodic waveforms and crossing frequencies. This a joint work with Clint Scovel (Caltech) and Gene Ryan Yoo (Caltech).