Machine Learning and Physics-based Modeling-Weinan E----力学研究所.pdf

机器学习与物理模型 Machine Learning and Physics-based Modeling: How can we construct interpretable and truly reliable physical models using concurrent machine learning? 鄂维南 Princeton University Joint work with: Jiequn Han, Han Wang, Linfeng Zhang Roberto Car, Chao Ma, Zheng Ma, Huan Lei, ...... May 14, 2020 1 / 57 Outline Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 2 / 57 PDEs and fundamental laws of physics Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 3 / 57 PDEs and fundamental laws of physics Two main themes of scientific research 寻求基本原理 Physics: Newton’s laws, Maxwell equations, Quantum mechanics 解决实际问题 “Engineering” (industrial) problems: 制造行业，材料，医疗行业。。。 May 14, 2020 4 / 57 PDEs and fundamental laws of physics 基本原理：Paul Dirac’s claim (1929) ”The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. ” 除少数物理领域（高能物理，核物理，天体物理）以外，对我们日常生活中碰到的问题，如化学, 材料, 生物, 工程领域中的问题，量子力学就足够了。困难的是，量子力学数学上太困难了。 i∂tΨ = HΨ, Ψ = Ψ(x1, x2, · · · , xN ) H= 系统的哈密顿算符 May 14, 2020 5 / 57 PDEs and fundamental laws of physics 怎样从基本原理出发解决实际问题？数值方法简化模型多尺度模型／算法 May 14, 2020 6 / 57 PDEs and fundamental laws of physics 简化模型对简化模型的要求： express fundamental physical principles (e.g. conservation laws) obey physical constraints (e.g. symmetries, frame-indifference, Galilean invariance) (universally) accurate (transferrable): physical parameters can be measured using simple experiments physically meaningful (interpretable) Very successful example: Euler’s equation for gas dynamics (for dense gas) A not very successful example: extended Euler equation for rarified gas (e.g. Grad’s 13-moment equation) 分子平均自由程 Kn = 系统尺度 May 14, 2020 7 / 57 PDEs and fundamental laws of physics 怎样简化模型？物理方法 generalized hydrodynamics: Onsager Landau: gradient expansion, weakly nonlinear theory successful example: Ericksen-Leslie equation for liquid crystals unsuccessful example: Non-Newtonian fluids 数学方法渐近分析, e.g. PLK (Poincaré-Lighthill-Kuo) method projection onto principal components truncation (e.g. Lorenz system) May 14, 2020 8 / 57 PDEs and fundamental laws of physics Lorenz system Drastically simplified model for 2D incompressible flow with buoyancy and gravity. x = ψ11 ∼ convective intensity y = T11 ∼ temperature difference between descending and ascending currents z = T02 ∼ difference in vertical temperature ẋ = σ(y − x), ẏ = −xz + rx − y, ż = xy − bz where σ = ν/κ, r = Ra/Rac. Question: Is this a good model for the original problem? May 14, 2020 9 / 57 PDEs and fundamental laws of physics 数值方法 finite difference finite element spectral methods ...... These have completely changed the way we do science, and to an even greater extend, engineering. gas dynamics structural analysis radar, sonar, optics control of flight vehicles, satellites ...... 低维问题基本解决，高维（或多个自由度）问题极度困难 Curse of dimensionality (CoD) 维数灾难: As the dimension grows, the complexity (or computational cost) grows exponentially. May 14, 2020 10 / 57 PDEs and fundamental laws of physics 多尺度方法/物理力学借助微观模型解决宏观问题, by performing (many) small scale microscale simulations (see Heterogeneous Multi-scale Method by E and Engquist). Difficulty: Microscale models are not very reliable either The scale of microscale simulation is still out of reach Lack of clear separation of scales May 14, 2020 11 / 57 PDEs and fundamental laws of physics Many problems remain Within mechanics: equations for solids? nonlinear elasticity, plasticity, visco-plasticity, crack propagation non-Newtonian fluids (generalized Newtonian, Maxwell models, co-rotational, co-deformational) turbulence: RANS (k − ε models), LES rarefied gas dynamics Outside of mechanics Density functional theory: Exchange-correlation functional Molecular dynamics: Potential energy surface (PES) Coarse-graind molecular dynamics: Free energy function In the absence of systematic methods, one has to resort to ad hoc approaches. May 14, 2020 12 / 57 PDEs and fundamental laws of physics Machine learning comes to the rescue Modern machine learning provides a way to solve the CoD problem. Two objectives: interpretable and truly reliable physical models with machine learning multi-scale modeling in situations without scale separation Remark: 机器学习不是解决困难问题的万能工具怎样用好机器学习是一个nontrivial problem May 14, 2020 13 / 57 Machine learning Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 14 / 57 Machine learning Basic example: Supervised learning Given S = {(xj , yj = f ∗(xj )), j ∈ [n]}, learn f ∗. 1 2 This is a problem about function approximation. Based on finite pieces of “labeled” data regression (f ∗ is continuous) vs classification (f ∗ is discrete) will neglect measurement noise (not crucial for the talk) assume xj ∈ X = [0, 1]d. d is typically quite large notation: µ = the distribution of {xj } In practice, one divides S into two subsets, a training set and a testing set. May 14, 2020 15 / 57 Machine learning Classification example: Cifar 10 Input: x ∈ [0, 1]d with d = 32 × 32 × 3 = 3072. Output: f ∗ ∈ {0, 1, . . . , 9}. May 14, 2020 16 / 57 Machine learning ML and approximation theory Classical approximation theory: polynomials, piece-wise polynomials, wavelets, splines curse of dimensionality: kf ∗ − fmk ∼ m−α/dΓ(f ∗) ML: high dimensionality May 14, 2020 17 / 57 Machine learning Dealing with high dimensionality: Monte Carlo integration Z I(g) = g(x)dµ, [0,1]d 1X In(g) = g(xj ) n j Trapezoidal rule: I(g) − In(g) ∼ m−α/dΓ(g) Monte Carlo: {xj , j ∈ [n]} is uniformly distributed in [0, 1]d. var(g) , E(I(g) − In(g)) = n 2 Z var(g) = g 2(x)dx − X 2 Z g(x)dx X √ The O(1/ n) rate is (almost) the best we can hope for. However, var(g) can be very large in high dimension. Variance reduction! May 14, 2020 18 / 57 Machine learning Representing functions: An illustrative example Traditional approach: Z f (x) = a(ω)e i(ω,x) dω, Rd 1 X fm(x) = a(ωj )ei(ωj ,x) m j {ωj } is a fixed grid, e.g. uniform. kf − fmkL2(X) ≤ C0m−α/dkf kH α(X) “New” approach: Z f (x) = Rd a(ω)ei(ω,x)π(dω) = Eω∼π a(ω)ei(ω,x) where π is a probability measure on Rd. Let {ωj } be an i.i.d. sample of π. m var(f ) 1 X i(ωj ,x) 2 E|f (x) − a(ωj )e | = m j=1 m where var(f ) = Eω∼π |a(ω)|2 − f (x)2 May 14, 2020 19 / 57 Machine learning Two types of machine learning models (1). Models that suffer from CoD: generalization error = O(m−α/d) and/or O(n−β/d) piecewise polynomial approximation wavelets with fixed wavelet basis (2). Models that don’t suffer from CoD: For example √ generalization error = O(γ1(f )/m + γ2(f )/ n) ∗ ∗ These are “Monte-Carlo-like” bounds, γ1 and γ2 play the role of variance in Monte Carlo. random feature models neural network models May 14, 2020 20 / 57 Concurrent learning Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 21 / 57 Concurrent learning Sequential vs concurrent learning Where are the data sets? (It is very expensive to get the data) sequential learning: first collect labeled data {xj , yj }, then perform learning concurrent learning: generate the data set on the fly as learning proceeds compare with ”active learning”: having unlabeled data {xj }, and decide which ones to label and use them to perform learning concurrent learning: generate “optimal data set” (both unlabeled and labeled, representative enough yet as small as possible) the latter is a more interactive process May 14, 2020 22 / 57 Concurrent learning The exploration-labeling-training (ELT) procedure for concurrent learning Zhang, Wang and E (2018), J. Chem. Phys. Start out with no (macro-scale) model, no data; but with a micro-scale model. Repeat the following steps: 1 exploration: explore the configuration space, and decide which configurations need to be labeled. 2 labeling: compute the micro-scale solutions for the configurations that need to be labeled. This is our data set. 3 training: train the macro-scale model, and use it to help the exploration Similar to “active learning” but more interactive...... May 14, 2020 23 / 57 Concurrent learning The ELT algorithm p Indicator: = maxi hkfi − f¯ik2i, f¯i = hfii May 14, 2020 24 / 57 Molecular modeling Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 25 / 57 Molecular modeling Molecular dynamics Traditional dilemma: accuracy vs cost. E = E(R1, R2, ..., Ri, ..., RN ), d2Ri mi 2 = Fi = −∇Ri E. dt Two ways to calculate E and F : Computing the inter-atomic forces on the fly using QM, e.g. the Car-Parrinello MD. Accurate but expensive: X KS KS Λij φj . E = hΨ0|He |Ψ0i, µφ̈i = He φi + j Empirical potentials: efficient but unreliable. The Lennard-Jones potential (rij = |Ri − Rj |): 12 6! σ σ 1X Vij = 4 − , E= Vij . rij rij 2 i6=j May 14, 2020 26 / 57 Molecular modeling Integrating ML with molecular modeling New paradigm: quantum mechanics model – data generator machine learning – parametrize (represent) the model molecular dynamics – simulator Important issues: How do we make sure that the data is good enough? How do we enforce physical constraints (symmetries)? May 14, 2020 27 / 57 Molecular modeling Deep Potential: construction Structure: composite neural networks (NNs). E = i E . i P T T T T Ri = {R1i , · · · , Rji , · · · , RN } , Rji = Rj − Ri. i ,i Models of this type are naturally extensive. May 14, 2020 28 / 57 Molecular modeling The importance of preserving the symmetries Symmetries: Translational, rotational and permutational May 14, 2020 29 / 57 Molecular modeling Preserving symmetry: Poor man’s version remove translational and rotational symmetry by fixing a local frame of reference remove permutational symmetry by fixing an ordering of the atoms in the neighborhood creates small discontinuity when atoms switch their orders. Figure: deep potential molecular dynamics (DPMD) May 14, 2020 30 / 57 Molecular modeling Preserving the symmetries Translation, rotation, and permutation. T̂bf (r) = f (r + b), R̂U f (r) = f (rU), P̂σ f (r) = f (rσ(1), rσ(2), ..., rσ(N )) Translation and Rotation: Ωijk = rji · rki. Lemma: Ωijk is an overcomplete array of basic invariants with respect to rotation, reflection, and translation. Permutation: X g(rji)rji. j∈N (i) Lemma: A function f (r1i , ..., rji , ..., rNi i ) is invariant to the permutation of instances in rji , if and only if it can be P decomposed in the form ρ( j∈N (i) g(rji )rji ), for suitable transformations g and ρ. May 14, 2020 31 / 57 Molecular modeling Deep Potential: smooth version The whole sub-network consists of an encoding net Di(Ri) and a fitting net E i(Di). (Rotation: R̃i(R̃i)T , permutation: (G i1)T R̃i and (R̃i)T G i2.) May 14, 2020 32 / 57 Molecular modeling DP-GEN: generating optimal dataset using concurrent learning The exploration-labeling-training procedure Exploration: Sample the (T, p) space For each value of (T, p), sample the canonical ensemble (using DPMD). In addition, initialize the exploration with a variety of different initial configurations. Labeling: Using DFT (with periodic boundary condition) Training: Using “Deep Potential” May 14, 2020 33 / 57 Molecular modeling Example: Al, Mg, Al-Mg ∼0.005% configurations explored are selected for labeling. May 14, 2020 34 / 57 Molecular modeling Case 1: accuracy is comparable to the accuracy of the data May 14, 2020 35 / 57 Molecular modeling Case 2: structural information of DFT water Radial and angular distribution function of liquid water (PI-AIMD): Distribution of the Steinhardt order parameter Q̄6: May 14, 2020 36 / 57 Molecular modeling DP-GEN for water P 0K 1 TPa 50 K 100 K 1GPa 1 GPa Temperature 300 K 350 K 400 K 450 K 500 K 550 K 600 K XI(hexagonal) X 650 K 10 Mbar XV IX V 218 K, 620 MPa 248.85 K, 344.3 MPa 238.5 K, 212.9 MPa Solid 100 kbar 278 K, 2.1 GPa 355.00 K, 2.216 GPa VI 272.99 K, 632.4 MPa 256.164 K, 350.1 MPa 251.165 K, 209.9 MPa III 10 kbar Critical point Liquid 10 bar XI Ic Ih (ortho- Freezing point at 1 atm 273.15 K, 101.325 kPa rhombic) 1 bar Boiling point at 1 atm 373.15 K, 101.325 kPa 100 mbar 1 kPa 10 mbar Solid/Liquid/Vapour triple point 273.16 K, 611.657 Pa 100 Pa 1 mbar Vapour 10 Pa 1Pa 1 kbar 100 bar 10 kPa 1KPa Ionic. Liq. 647 K, 22.064 MPa 1 MPa 100 kPa 2500K SI 1 Mbar VII II 10 MPa Pressure 250 K VIII 100 MPa 1MPa 200 K 100 K, 62 GPa 100 GPa 10 GPa 150 K 1 Pa 100 µbar -250 °C -200 °C -150 °C -100 °C 0K -50 °C 0 °C 50 °C 100 °C 150 °C 200 °C 250 °C 300 °C 350 °C 10 µbar 200K 400K 600K T Reference model: DFT at the classical SCAN level; Starting configurations: relaxed Ice I-XV at T = 0 K and equilibrated liquid at T = 330 K; Range of thermodynamic conditions: red dashed box; number of MD snapshots: DPMD exploration: 1.4 billion, DFT calculation: 32 thousand (∼0.002% of the former). Typical AIMD trajectory: 100 thousand snapshots (50-100 ps). number of DP-GEN iterations: 100. May 14, 2020 37 / 57 Molecular modeling Lithium diffusion in solid-state electrolyte Ability to handle multi-component systems, here the LiGePS-type systems. Jun Cheng’s group at Xiamen U May 14, 2020 38 / 57 Molecular modeling 86 PFLOPS DeePMD simulation of 100M atoms D. Lu, et al, arXiv: 2004.11658; W. Jia, et al, arXiv: 2005.00223 May 14, 2020 39 / 57 Molecular modeling Open-source softwares: DeePMD-kit TensorFlow: efficient network operators LAMMPS, i-PI; MPI/GPU support. Free download from https://github.com/deepmodeling/deepmd-kit H. Wang,. et al, .Comp.Phys.Comm., 0010-4655 (2018). May 14, 2020 40 / 57 Molecular modeling Open-source softwares: DP-GEN Free download from https://github.com/deepmodeling/dpgen May 14, 2020 41 / 57 Molecular modeling Discussion group bbs.deepmd.org May 14, 2020 42 / 57 Molecular modeling 1 physical/chemical problems understanding water (phase diagram of water, including reactive regions; phase transition: ice to water, ionic liquid to super-ionic ice; nuclear quantum effect: collective tunneling, isotope effect; reactive event: dissociation and recombination; water surface and water/TiO2 interface; spectra: infra-red; Raman; X-ray Absorption; exotic properties: dielectric constant; density anomaly, etc. physical understanding of different systems that require long-time large-scale simulation with high degrees of model fidelity ( high-pressure iron: fractional defect; phase boundary; high-pressure hydrogen: exotic phases) catalysis (Pt cluster on MoS2 surface; CO molecules on gold surface, etc.) 2 materials science problems battery materials (diffusion of lithium in LGePS, LSGeSiPS, etc.; diffusion of Se in Cu2Se alloy) high entropy/high temperature alloy (CoCrFeMnNi alloy; Ni-based alloy) 3 organic chemistry/bio problems crystal structure prediction of molecular crystals; protein-ligand interaction; protein folding. May 14, 2020 43 / 57 Kinetic model for gas dynamics Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 44 / 57 Kinetic model for gas dynamics Modeling gas dynamics mean free path Kn = macroscopic length NSF Eqn ½ 10¡2 10¡1 ! equilibrium ! ! non-equilibrium ! 1.0 free flight ½ kinetic regime Kn ½ Euler Eqn transition regime ½ 1 CONTINUUM M ½ me History and Background 10.0 ure 1: Overview of the range of Knudsen number and various model regim May 14, 2020 45 / 57 Kinetic model for gas dynamics Boltzmann Equation One-particle density function f (x, v, t) 1 ∂tf + v · ∇xf = Q(f ), ε v ∈ R3 , x ∈ Ω ⊂ R3 , ε = Kn = Knudsen number and Q is the collision operator. Macroscopic state variables: ρ, u and T (density, bulk velocity and temperature) Z Z Z 1 1 f v dv, T = f |v − u|2 dv. ρ = f dv, u = ρ 3ρ When ε 1, Boltzmann can be approximated by Euler: ∂tU + ∇x · F (U ) = 0, with p = ρT , E = 12 ρu2 + 32 ρT , U = (ρ, ρu, E)T F (U ) = (ρu, ρu ⊗ u + pI, (E + p)u)T May 14, 2020 46 / 57 Kinetic model for gas dynamics Conventional Moment Method Proceed in 3 steps: 1. Start with the choice of a finite-dimensional linear subspace of functions of v (usually to be polynomials, e.g., Hermite polynomials). 2. Expand f (x, v, t) using these functions as bases and take the coefficients as moments (including macroscopic variables ρ, u, T , etc.). 3. Finally close the system with simplified assumptions, e.g., truncating moments of higher orders  ∂tU + ∇x · F (U , W ) = 0, 1 ∂tW + ∇x · G(U , W ) = R(U , W ). ε R R G(U , bW ) ∼ vw(v)f dv, R(U , W ) ∼ Q(v)w(v)dv For instance, in Grad 13-moment system, (U , W ) is constructed based on the moments of the bases {1, v, (v − u) ⊗ (v − u), |v − u|2(v − u)}. May 14, 2020 47 / 57 Kinetic model for gas dynamics Machine learning-based moment method Objective: construct an uniformly accurate (generalized) moment model using machine learning. 1: Learn the Moments through Autoencoder Find an encoder Ψ that maps f (·, v) to generalized moments W ∈ RM and a decoder Φ that recovers the original f from U , W Z W = Ψ(f ) = wf dv, Φ(U , W )(v) = h(v; U , W ). The goal is essentially to find optimal w and h parametrized by neural networks through minimizing E kf − Φ(Ψ(f ))k2 + λη (η(f ) − hη (U , W ))2. f ∼D η(f ) denotes entropy. May 14, 2020 48 / 57 Kinetic model for gas dynamics 2: Learn the Fluxes and Source Terms in the PDE Recall the general conservative form of the moment system ∂tU + ∇x · F (U , W ; ε) = 0, ∂tW + ∇x · G(U , W ; ε) = R(U , W ; ε). Rewrite it into (variance reduction) ( ∂tU + ∇x · [F0(U ) + F̃ (U , W ; ε)] = 0, ∂tW + ∇x · [G0(U ) + G̃(U , W ; ε)] = R(U , W ; ε). F0(U ), G0(U ) are the fluxes of the moments U , W under the Maxwellian distribution. Our goal is to obtain ML models for F̃ , G̃, R from the original kinetic equation. Issues: (1) physical symmetries (e.g. Galilean invariance); (2) data generation (active learning algorithm); (3) locality vs. non-locality of the model May 14, 2020 49 / 57 Kinetic model for gas dynamics The ELT algorithm 𝑈 𝑊 preparing the data ﬁnding the moments learning the closure exploring the data Figure: Schematic diagram of the machine learning-based moment method exploration: random initial conditions made up of waves and discontinuities labeling: solving kinetic equation (Boltzmann equation for Maxwell molecules) training: Galilean invariance May 14, 2020 50 / 57 Kinetic model for gas dynamics Galilean Invariant Moments Galilean invariance of the Boltzmann equation: f 0(x, u, t) = f (x − tu0, v − u0, t). Moments: Z WGal = Ψ(f ) = v−u dv. f (v)w √ T Closure: Z ∂t f (v)w RD v − uj p Tj ! Z dv + ∇x · f (v)w RD v − uj p Tj ! v T dv = Z 1 Q(f )w RD ε v − uj p Tj ! dv. 1 ∂tWGal + ∇x · GGal(U , WGal; Uj ) = RGal(U , WGal). ε The data efficiency is better than the previous one since it learns the dynamical system more intrinsically. May 14, 2020 51 / 57 Kinetic model for gas dynamics ε ∼ Log10-Uniform(−3, 1), constant across the domain; initial profiles consist of a combination of a few sin waves and shocks. Size of dataset array: 200 × 100 × 48 × 48 × 100. Specify W ∈ R9. 1.2 0.04 1.0 0.8 1.0 0.02 Energy Momentum Density 1.2 0.00 −0.02 0.6 −0.04 0.6 x x 0.3 1.1 1.1 0.2 1.0 1.0 0.9 0.8 0.7 Energy 1.2 Momentum Density x 0.1 0.0 −0.1 0.9 0.8 0.7 0.6 −0.2 0.6 0.5 x x x 0.3 1.2 1.1 1.0 0.9 0.8 0.2 1.0 0.1 0.9 Energy Momentum 1.1 Density 0.8 0.0 0.8 0.7 0.7 −0.1 0.6 0.6 −0.2 0.5 −0.4 −0.2 0.0 x 0.2 0.4 −0.4 Boltzmann −0.2 0.0 0.2 0.4 x Euler −0.4 −0.2 0.0 0.2 0.4 x GalEncMLC Figure: Sample profiles of ρ, ρu, E (from left to right) at t = 0, 0.05, 0.1 (from top to bottom), ε = 8.10 May 14, 2020 52 / 57 Kinetic model for gas dynamics ε varies from 10−3 to 10 in the domain; initial profiles are the same as before 1.05 1.4 0.04 0.95 0.90 0.85 0.80 0.75 1.2 0.02 Energy Momentum Density 1.00 0.00 −0.02 0.8 0.6 −0.04 0.70 1.0 0.4 x x x 1.05 1.4 0.4 1.00 0.90 0.85 0.80 0.75 0.70 0.0 −0.2 1.1 0.4 x 1.4 0.8 1.2 0.2 Energy Momentum 0.9 0.8 x 0.4 1.0 1.0 0.6 −0.4 x Density 1.2 0.2 Energy Momentum Density 0.95 0.0 −0.2 1.0 0.8 0.6 0.7 −0.4 −0.2 0.0 x 0.2 0.4 −0.4 0.4 −0.4 Boltzmann −0.2 0.0 0.2 0.4 x Euler −0.4 −0.2 0.0 0.2 0.4 x GalEncMLC Figure: Profiles of ρ, ρu, E (from left to right) at t = 0, 0.05, 0.1 (from top to bottom) May 14, 2020 53 / 57 Kinetic model for gas dynamics Numerical results Learned functions w(v) as generalized moments −2 3 4 2 w3 w2 w1 0 5 3 0 2 −4 −1 1 −10 −5 0 v 5 −10 10 1 −5 0 v 5 10 −10 −5 0 v 5 10 −10 −5 0 v 5 10 4 2 −2 −3 0 −4 −2 −5 −10 −5 0 v 5 10 w6 w5 w4 2 0 −2 −10 −5 0 v 5 10 −4 May 14, 2020 54 / 57 Concluding remarks Outline 1 PDEs and fundamental laws of physics 2 Machine learning 3 Concurrent learning 4 Molecular modeling 5 Kinetic model for gas dynamics 6 Concluding remarks May 14, 2020 55 / 57 Concluding remarks Concluding remarks The integration of machine learning with physics-based modeling opens up a whole new chapter in the way we do scientific and engineering modeling Concurrent learning ensures generation of the optimal dataset It is important to take “physics” into account These are new physical models, not just algorithms May 14, 2020 56 / 57