Materials Process Design and Control Laboratory An information-learning approach for multiscale modeling of materials Sethuraman Sankaran and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL:

Materials Process Design and Control Laboratory Research Sponsors U.S. AIR FORCE PARTNERS Materials Process Design Branch, AFRL Computational Mathematics Program, AFOSR CORNELL THEORY CENTER ARMY RESEARCH OFFICE Mechanical Behavior of Materials Program NATIONAL SCIENCE FOUNDATION (NSF) Design and Integration Engineering Program

Materials Process Design and Control Laboratory Why do we need a statistical model of microstructures? When a specimen is manufactured, the microstructures at a sample point will not be the same always. How do we compute the class of microstructures based on some limited information? Different statistical samples of the manufactured specimen

Materials Process Design and Control Laboratory Need for multiscaling Concept of multiscaling Microstructural randomness at a sample point in different gear specimens Technique Limited microstructural information: grain size moments and average textures Train learning tool based on information measures Compute microstructural variability in grain sizes and ODFs (texture) Compute variation in homogenized plastic non-linear properties Micro Macro MaxEnt

Materials Process Design and Control Laboratory Phase field/experimental microstructures Derive a learning machine using information measures Generate PDF of microstructures Compute property bounds for the class of microstructures Macro scale Meso scale Macro scale Information Learning Maximum entropy technique Homogenization Main idea

Materials Process Design and Control Laboratory Generating input microstructures: The phase field model Define order parameters: where Q is the total number of orientations possible Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) : Non-zero only near grain boundaries

Materials Process Design and Control Laboratory Physics of phase field method Driving force for grain growth:  Reduction in free energy: thermodynamic driving force to eliminate grain boundary area (Ginzburg-Landau equations) kinetic rate coefficients related to the mobility of grain boundaries Assumption: Grain boundary mobilties are constant

Materials Process Design and Control Laboratory Phase Field – Problem parameters Isotropic mobility (L=1) Isotropic mobility (L=1) Discretization : Discretization : problem size : 75x75x75 Order parameters: Q=20 Timesteps = 1000 Timesteps = 1000 First nearest neighbor approx. First nearest neighbor approx.

Materials Process Design and Control Laboratory Phase field/experimental microstructures Derive a learning machine using information measures Generate PDF of microstructures Compute property bounds for the class of microstructures Macro scale Meso scale Macro scale Information Learning Maximum entropy technique Homogenization Main idea Extract average and lower grain size moments

Materials Process Design and Control Laboratory The MAXENT principle The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown. E.T. Jaynes 1957 MAXENT is a guiding principle to construct PDFs based on limited information There is no proof behind the MAXENT principle. The intuition for choosing distribution with maximum entropy is derived from several diverse natural phenomenon and it works in practice. The missing information in the input data is fit into a probabilistic model such that randomness induced by the missing data is maximized. This step minimizes assumptions about unknown information about the system.

Materials Process Design and Control Laboratory Subject to Lagrange Multiplier optimization feature constraints features of image I MAXENT as an optimization problem Partition Function Find

Materials Process Design and Control Laboratory Gradient Evaluation Objective function and its gradients: Objective function and its gradients: Infeasible to compute at all points in one conjugate gradient iteration Infeasible to compute at all points in one conjugate gradient iteration Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler) Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler)

Materials Process Design and Control Laboratory Gibbs Sampler Integration over extremely large dimensional space. Very time consuming if this is to be done at each iteration Solution: Learn the integral for certain values of the parameter using information learning Use the parameters of the learning machine to compute integrals Maximum entropy approach for statistical modeling of three dimensional polycrystal microstructures, Mathematical and Computational aspects of multi-scale and multi- physics. Thursday, 16:40 pm. Santa Monica Room

Materials Process Design and Control Laboratory Phase field/experimental microstructures Derive a learning machine using information measures Generate PDF of microstructures Compute property bounds for the class of microstructures Macro scale Meso scale Macro scale Information Learning Maximum entropy technique Homogenization Extract average and lower grain size moments

Materials Process Design and Control Laboratory Information Learning The goal of learning is to optimize the performance of a parametric mapper according to some cost function In classification, minimize the probability of error In regression the goal is to minimize the error in the fit (goal of this talk) Information Learning is an information-theoretic scheme that tries to reconstruct a process from a set of data based purely on information measures. Information Learning P Y X Computed as input parameters of Gibbs sampling algorithm Lower moments of grain sizes

Materials Process Design and Control Laboratory Minimize error entropy or maximize the mutual information between actual and desired output. Information Learning scheme Classical Information Learning System Learning System Y=WH(X,s) The scheme is based on interactions between pairs of information particles.

Materials Process Design and Control Laboratory Learning based on phase field simulations Why do we need information learning? Microstructures obtained from experiments or simulations tend to have highly heterogeneous grain sizes Features such as grain sizes are statistical in nature Deterministic problems: Neural network Statistical problems: Information Learning

Materials Process Design and Control Laboratory Theory: quantification of inputs Shannon’s Entropy Renyi’s Entropy Shannon is a special case when Quadratic entropy: ( ) Define: V: Information potential

Materials Process Design and Control Laboratory Training information networks: back-propagation algorithm Learning rate Error entropy Suppose f e is gaussian, Computed using standard back-propagation algorithm

Materials Process Design and Control Laboratory Information potential and force After some computations, we get Information force on a particle: Force between two information-particles. Higher the force, the weights are farther away from equilibrium

Materials Process Design and Control Laboratory Main back-propagation algorithm When the error is small w.r.t. the kernel size, quadratic entropy training is equivalent to a biased MSE. Hence, the real advantage of entropy training is for NONLINEAR systems. The algorithm is O(N 2 ).

Materials Process Design and Control Laboratory LM optimisation for optimization of E=MSE or EE(error entropy). Requires first and second derivatives of the objective function (E). Parameters are updated by Increase to improve stability (at expense of decreasing speed of convergence). Optimization algorithms Backpropagation Parameters are updated by decides the rate of convergence

Materials Process Design and Control Laboratory Phase field/experimental microstructures Derive a learning machine using information measures Generate PDF of microstructures Compute property bounds for the class of microstructures Macro scale Meso scale Macro scale Information Learning Maximum entropy technique Homogenization Main idea

Materials Process Design and Control Laboratory (First order) homogenization scheme 1.Microstructure is a representation of a material point at a smaller scale 2.Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)Hill Design of microstructures in polycrystals using multi-scale homogenization techniques, Multiscale computational design of products and materials. Tuesday, 15:48 pm. Westwood Room

Materials Process Design and Control Laboratory Numerical examples

Materials Process Design and Control Laboratory Learning grain size information from phase field simulations Problem definition: Using phase-field simulations, samples of microstructures are generated. Randomness in grain sizes of resulting microstructures result due to randomness in (i) grain nucleation sites and (ii) randomness in mobility. Information-learning systems utilize grain sizes as a feature to compute parametric weights of the non-linear mapper.

Materials Process Design and Control Laboratory Three characteristic microstructures computed using phase-field simulations. Computations involved a set of 36 phase field variables with random mobilities. A set of FIVE input microstructures were utilized for the problem. Samples from phase field simulation of microstructures

Materials Process Design and Control Laboratory Convergence of the learning tool: grain size Three moments of grain sizes were utilized to train an information-learning tool. The convergence of the mean error estimate as well as the information potential are shown in the figure Iterations L2 error norm Iterations Information potential Mean square error Information Potential

Materials Process Design and Control Laboratory Comparison of grain size distributions The figure shows a comparison of grain size distribution that is computed using the information learning machine as well as that generated using the input set of microstructures Gibbs sampler: 1 iteration = 0.26 cpu clock seconds Information network: 1 iteration = 0.01 cpu clock seconds

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL= Reconstructing microstructures from Learning database Computing microstructures using the Information Learning method

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL=0.05 Computing microstructures using the Information Learning method Reconstructing microstructures from Learning database

Materials Process Design and Control Laboratory Texture features of the learning tool: ODF Crystal/lattice reference frame e2e2 ^ Sample reference frame e1e1 ^ e’ 1 ^ e’ 2 ^crystal e’ 3 ^ e3e3 ^  ORIENTATION SPACE Euler angles – symmetries Neo Eulerian representation n Rodrigues’ parametrization Average ODF is known. We need to compute a distribution of ODF’s Use Gibbs sampler and train the information network for the same

Materials Process Design and Control Laboratory Error features of the learning tool: ODF A sample of hundred ODFs (orientation distribution functions) drawn from a class of ODFs are used to train a system. The computed error norms and information potentials based on a non-linear information-theoretic mapper are shown in the figure Iterations L2 error norm Iterations Information Potential

Materials Process Design and Control Laboratory Computed textural features based on the learning tool Orientation angle (in radians) Orientation distribution function ODF computed based on input data An ODF sample computed from learning machine

Materials Process Design and Control Laboratory Statistical variation of properties Statistical variation of homogenized stress- strain curves. Aluminium polycrystal with rate-independent strain hardening. Pure tensile test.

Materials Process Design and Control Laboratory Conclusions and Future Work

Materials Process Design and Control Laboratory Information learning is an extremely useful tool for storing and generating statistical data that occurs due to randomness in grain sizes and textures of microstructures Information learning is an extremely useful tool for storing and generating statistical data that occurs due to randomness in grain sizes and textures of microstructures Numerical examples depicting how data can be stored using information learning technique and extracted features from the learning tool was shown to produce consistent results Numerical examples depicting how data can be stored using information learning technique and extracted features from the learning tool was shown to produce consistent results The information network can also be used to compute distributions of grain sizes and orientations. The information network can also be used to compute distributions of grain sizes and orientations. Conclusions

Materials Process Design and Control Laboratory Use data generated from phase field simulations for crystal growth and dendritic solidification to generate a non-linear mapper computed using the information learning scheme Future application: Database for dendritic solidification