Data-Driven Computational Mechanics (II)
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1 C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y Data-Driven Computational Mechanics (II) California Institute of Technology and Rheinische Friedrich-Wilhelms Universität Bonn COMPLAS 2017 Barcelona, September 5, 2017
2 Data Science, Big Data What s in it for us?
3 Data Science, Big Data What s in it for us? Data Science is the extraction of knowledge from large volumes of unstructured data 1 Data science requires sorting through big-data sets and extracting insights from these data Data science uses data management, statistics and machine learning to derive mathematical models for subsequent use in decision making Data science influences (non-stem!) fields such as marketing, advertising, finance, social sciences, security, policy, medical informatics But What s in it for us? 1 Dhar, V., Communications of the ACM, 56(12) (2013) p. 64.
4 Where is Data Science needed in Computational Mechanics?
5 Where is Data Science needed in Computational Mechanics? Universal laws! (Newton s laws, Schrodinger s eq., Maxwell s eqs., Einstein s eqs ) Exactly known! Uncertainty-free! (epistemic)
6 Where is Data Science needed in Computational Mechanics? Unknown! Epistemic uncertainty!
7 Data Science and material modeling Need to generate (epistemic) knowledge about material behavior to close BV problems Traditional modeling paradigm: Fit data (a.k.a. regression, machine learning, model reduction, central manifolds ), use calibrated empirical models in BV problems
8 Data Science and the classical Modeling & Simulation paradigm Material data Modeling funnel Material model funnel Simulation Manufactured data broken pipe!
9 Data Science and material modeling Need to generate (epistemic) knowledge about material behavior to close BV problems Traditional modeling paradigm: Fit data (a.k.a. regression, machine learning, model reduction, central manifolds ), use calibrated empirical models in BV problems But: We live in a data-rich world (full-field diagnostics, data mining from first principles ) Data-Driven paradigm: Use material data directly in BV (no fitting by any name, no loss of information, no broken pipe between material data and manufactured data) How? (math talks, nonsense walks )
10 Elementary example: Bar and spring Phase space
11 Elementary example: Bar and spring Phase space
12 The general Data-Driven (DD) problem The Data-Driven problem 1 : Given, D = {material data}, E = {compatibility + equilibrium}, The aim Data-Driven analysis is to find the compatible and equilibrated solution that is closest to the material data set No material modeling, no data fitting, no V&V Raw material data is used (unprocessed) in calculations (the facts, nothing but the facts ) Are Data-Driven problems well-posed? 1 T. Kirchdoerfer and M. Ortiz (2015) arxiv: T. Kirchdoerfer and M. Ortiz, CMAME, 304 (2016)
13 Data-Driven elasticity σσ εε S. Conti, S. Müller and M. Ortiz (2017) arxiv:1708:02880
14 Data-Driven elasticity Well-posedness S. Conti, S. Müller and M. Ortiz (2017) arxiv:1708:02880
15 Data-Driven elasticity Convergence with respect to the data set S. Conti, S. Müller and M. Ortiz (2017) arxiv:1708:02880
16 Data-Driven elasticity - Relaxation εε σσ μμ 2 μμ 1 εε 1 εε 2 S. Conti, S. Müller and M. Ortiz (2017) arxiv:1708:02880
17 Data-Driven Problems Data-driven problems represent a complete reformulation of the classical problems of mechanics (data + differential constraints) Data-driven problems subsume and are strictly larger than classical problems Data-driven analysis leads to notions of convergence of data sets that imply convergence of solutions. Data-driven relaxation (micro-macro) is fundamentally different from classical relaxation of energy functions!
18 Implementation: Trusses
19 Implementation: Trusses Standard linear truss problem Search over material-data set
20 Implementation: Staggered solver
21 Truss example: Convergence of solver Material-data sets of increasing size and decreasing scatter Convergence, local data assignment iteration
22 Truss example: Convergence wrt data Material-data sets of increasing size and decreasing scatter Convergence with respect to sample size (with initial Gaussian noise)
23 Distance-based DD solvers Distance-based DD solvers exhibit good convergence wrt to material data associations Distance-based DD solvers exhibit good convergence wrt uniformly converging data But distance-based DD solvers can be overly sensitive to outliers in the data (non-uniform data convergence) If outliers cannot be ruled out, distance-based DD solvers need to be generalized and extended
24 Extension to noisy data sets Phase space
25 Extension to noisy data sets Phase space Outlier!
26 Extension to noisy data sets Distance-based DD suffers from a tyranny of the outliers (non-uniform convergence) Eliminate by polling the data set more widely (cluster analysis, max-ent inference ) Thermalize distance to material set DD = (zz 1,, zz NN ): Max-ent DD problem 1 : Solve by simulated annealing! 1 T. Kirchdoerfer and M. Ortiz (2017) arxiv: v2
27 Extension to noisy data sets Phase space Outlier!
28 Truss example: Convergence wrt data Material-data sets of increasing size and decreasing scatter Convergence with respect to sample size (with Gaussian noise)
29 Extension to dynamics Constraint set: Time-discrete eqs. of motion (x direction) 1 T. Kirchdoerfer and M. Ortiz (2017) arxiv: v2
30 Concluding remarks Data-driven computing is emerging as an alternative paradigm to model-based computing Data-driven computing can reliably supply solutions from raw material data sets Data-driven computing is likely to be a growth area in an increasingly data-rich world Numerous outstanding questions: Phase-space coverage, importance sampling Building goal-oriented material data bases from experiment 1 and from first-principles calculations Inelasticity, path dependence 1 A. Leygue et al. (2017) HAL Id: hal
31 Concluding remarks
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