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Brunton SL, Kutz JN. Promising directions of machine learning for partial differential equations. NATURE COMPUTATIONAL SCIENCE 2024; 4:483-494. [PMID: 38942926 DOI: 10.1038/s43588-024-00643-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2023] [Accepted: 05/13/2024] [Indexed: 06/30/2024]
Abstract
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multiscale physics in a compact and symbolic representation. Here, we examine several promising avenues of PDE research that are being advanced by machine learning, including (1) discovering new governing PDEs and coarse-grained approximations for complex natural and engineered systems, (2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and (3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.
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Affiliation(s)
- Steven L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA.
| | - J Nathan Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA, USA
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Sahimi M. Physics-informed and data-driven discovery of governing equations for complex phenomena in heterogeneous media. Phys Rev E 2024; 109:041001. [PMID: 38755895 DOI: 10.1103/physreve.109.041001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2023] [Indexed: 05/18/2024]
Abstract
Rapid evolution of sensor technology, advances in instrumentation, and progress in devising data-acquisition software and hardware are providing vast amounts of data for various complex phenomena that occur in heterogeneous media, ranging from those in atmospheric environment, to large-scale porous formations, and biological systems. The tremendous increase in the speed of scientific computing has also made it possible to emulate diverse multiscale and multiphysics phenomena that contain elements of stochasticity or heterogeneity, and to generate large volumes of numerical data for them. Thus, given a heterogeneous system with annealed or quenched disorder in which a complex phenomenon occurs, how should one analyze and model the system and phenomenon, explain the data, and make predictions for length and time scales much larger than those over which the data were collected? We divide such systems into three distinct classes. (i) Those for which the governing equations for the physical phenomena of interest, as well as data, are known, but solving the equations over large length scales and long times is very difficult. (ii) Those for which data are available, but the governing equations are only partially known, in the sense that they either contain various coefficients that must be evaluated based on the data, or that the number of degrees of freedom of the system is so large that deriving the complete equations is very difficult, if not impossible, as a result of which one must develop the governing equations with reduced dimensionality. (iii) In the third class are systems for which large amounts of data are available, but the governing equations for the phenomena of interest are not known. Several classes of physics-informed and data-driven approaches for analyzing and modeling of the three classes of systems have been emerging, which are based on machine learning, symbolic regression, the Koopman operator, the Mori-Zwanzig projection operator formulation, sparse identification of nonlinear dynamics, data assimilation combined with a neural network, and stochastic optimization and analysis. This perspective describes such methods and the latest developments in this highly important and rapidly expanding area and discusses possible future directions.
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Affiliation(s)
- Muhammad Sahimi
- Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA
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Glenn K, He J, Rochlin R, Teng S, Hecker JG, Novosselov I. Assessment of aerosol persistence in ICUs via low-cost sensor network and zonal models. Sci Rep 2023; 13:3992. [PMID: 36899063 PMCID: PMC10006437 DOI: 10.1038/s41598-023-30778-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2022] [Accepted: 03/01/2023] [Indexed: 03/12/2023] Open
Abstract
The COVID-19 pandemic raised public awareness about airborne particulate matter (PM) due to the spread of infectious diseases via the respiratory route. The persistence of potentially infectious aerosols in public spaces and the spread of nosocomial infections in medical settings deserve careful investigation; however, a systematic approach characterizing the fate of aerosols in clinical environments has not been reported. This paper presents a methodology for mapping aerosol propagation using a low-cost PM sensor network in ICU and adjacent environments and the subsequent development of the data-driven zonal model. Mimicking aerosol generation by a patient, we generated trace NaCl aerosols and monitored their propagation in the environment. In positive (closed door) and neutral-pressure (open door) ICUs, up to 6% or 19%, respectively, of all PM escaped through the door gaps; however, the outside sensors did not register an aerosol spike in negative-pressure ICUs. The K-means clustering analysis of temporospatial aerosol concentration data suggests that ICU can be represented by three distinct zones: (1) near the aerosol source, (2) room periphery, and (3) outside the room. The data suggests two-phase plume behavior: dispersion of the original aerosol spike throughout the room, followed by an evacuation phase where "well-mixed" aerosol concentration decayed uniformly. Decay rates were calculated for positive, neutral, and negative pressure operations, with negative-pressure rooms clearing out nearly twice as fast. These decay trends closely followed the air exchange rates. This research demonstrates the methodology for aerosol monitoring in medical settings. This study is limited by a relatively small data set and is specific to single-occupancy ICU rooms. Future work needs to evaluate medical settings with high risks of infectious disease transmission.
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Affiliation(s)
- K Glenn
- Department of Mechanical Engineering, University of Washington, Seattle, USA
| | - J He
- Department of Mechanical Engineering, University of Washington, Seattle, USA
| | - R Rochlin
- Department of Mechanical Engineering, University of Washington, Seattle, USA
| | - S Teng
- Department of Mechanical Engineering, University of Washington, Seattle, USA
| | - J G Hecker
- Department of Anesthesiology and Pain Medicine, University of Washington, Seattle, USA
| | - I Novosselov
- Department of Mechanical Engineering, University of Washington, Seattle, USA.
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Baddoo PJ, Herrmann B, McKeon BJ, Nathan Kutz J, Brunton SL. Physics-informed dynamic mode decomposition. Proc Math Phys Eng Sci 2023. [DOI: 10.1098/rspa.2022.0576] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/05/2023] Open
Abstract
In this work, we demonstrate how physical principles—such as symmetries, invariances and conservation laws—can be integrated into the
dynamic mode decomposition
(DMD). DMD is a widely used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD can produce models that are sensitive to noise, fail to generalize outside the training data and violate basic physical laws. Our physics-informed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles—conservation, self-adjointness, localization, causality and shift-equivariance—and derive several closed-form solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of problems, including energy-preserving fluid flow, the Schrödinger equation, solute advection-diffusion and three-dimensional transitional channel flow. In each case, piDMD outperforms standard DMD algorithms in metrics such as spectral identification, state prediction and estimation of optimal forcings and responses.
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Affiliation(s)
- Peter J. Baddoo
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Benjamin Herrmann
- Department of Mechanical Engineering, University of Chile, Beauchef 851, Santiago, Chile
| | - Beverley J. McKeon
- Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
| | - J. Nathan Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
| | - Steven L. Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
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The effects of reaction kinetics upon the instabilities in cathodic electrodeposition. Curr Opin Colloid Interface Sci 2022. [DOI: 10.1016/j.cocis.2022.101591] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Vinuesa R, Brunton SL. Enhancing computational fluid dynamics with machine learning. NATURE COMPUTATIONAL SCIENCE 2022; 2:358-366. [PMID: 38177587 DOI: 10.1038/s43588-022-00264-7] [Citation(s) in RCA: 14] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 05/17/2022] [Indexed: 01/06/2024]
Abstract
Machine learning is rapidly becoming a core technology for scientific computing, with numerous opportunities to advance the field of computational fluid dynamics. Here we highlight some of the areas of highest potential impact, including to accelerate direct numerical simulations, to improve turbulence closure modeling and to develop enhanced reduced-order models. We also discuss emerging areas of machine learning that are promising for computational fluid dynamics, as well as some potential limitations that should be taken into account.
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Affiliation(s)
- Ricardo Vinuesa
- FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden.
- Swedish e-Science Research Centre (SeRC), Stockholm, Sweden.
| | - Steven L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA
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Fasel U, Kutz JN, Brunton BW, Brunton SL. Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proc Math Phys Eng Sci 2022; 478:20210904. [PMID: 35450025 PMCID: PMC9006119 DOI: 10.1098/rspa.2021.0904] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 03/10/2022] [Indexed: 12/17/2022] Open
Abstract
Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of the nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world datasets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900 to 1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.
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Affiliation(s)
- U Fasel
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA
| | - J N Kutz
- Department of Applied Mathematics, University of Washington, Seattle, WA, USA
| | - B W Brunton
- Department of Biology, University of Washington, Seattle, WA, USA
| | - S L Brunton
- Department of Mechanical Engineering, University of Washington, Seattle, WA, USA
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