Cory Braker Scott

Cory Braker Scott

This preliminary paper presents initial explorations in rendering Iterated Function System (IFS) fractals using a differentiable rendering pipeline. Differentiable rendering is a recent innovation at the intersection of computer graphics and machine learning. A fractal rendering pipeline composed of differentiable operations opens up many possibilities for generating fractals that meet particular criteria. In this paper I demonstrate this pipeline by generating IFS fractals with fixed points that resemble a given target image - a famous problem known as the \emph{inverse IFS problem}. The main contributions of this work are as follows: 1) I demonstrate (and make code available) this rendering pipeline; 2) I discuss some of the nuances and pitfalls in gradient-descent-based optimization over fractal structures; 3) I discuss best practices to address some of these pitfalls; and finally 4) I discuss directions for further experiments to validate the technique.

Eric Mjolsness, Cory B. Scott

Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level number) that grow in a hierarchical fashion, so that: (1) the number of graph vertices and edges increases exponentially in level number; (2) bipartite graphs connect successive levels within a graph lineage and, as in multigrid methods, can constrain matrices relating successive levels; (3) using prolongation maps within a graph lineage, process-derived distance measures between graphs at successive levels can be defined; (4) a category of "graded graphs" can be defined, and using it low-cost "skeletal" variants of standard algebraic graph operations and type constructors (cross product, box product, disjoint sum, and function types) can be derived for graded graphs and hence hierarchical graph lineages; (5) these skeletal binary operators have similar but not identical algebraic and category-theoretic properties to their standard counterparts; (6) graph lineages and their skeletal product constructors can approach continuum limit objects. Additional space-efficient unary operators on graded graphs are also derived: thickening, which creates a graph lineage of multiscale graphs, and escalation to a graph lineage of search frontiers (useful as a generalization of adaptive grids and in defining "skeletal" functions). The result is an algebraic type theory for graded graphs and (hierarchical) graph lineages. The approach is expected to be well suited to defining hierarchical model architectures - "hierarchitectures" - and local sampling, search, or optimization algorithms on them. We demonstrate such application to deep neural networks (including visual and feature scale spaces) and to multigrid numerical methods.

Cory Braker Scott, Eric Mjolsness, Diane Oyen et al.

We present a method for learning "spectrally descriptive" edge weights for graphs. We generalize a previously known distance measure on graphs (Graph Diffusion Distance), thereby allowing it to be tuned to minimize an arbitrary loss function. Because all steps involved in calculating this modified GDD are differentiable, we demonstrate that it is possible for a small neural network model to learn edge weights which minimize loss. GDD alone does not effectively discriminate between graphs constructed from shoot apical meristem images of wild-type vs. mutant \emph{Arabidopsis thaliana} specimens. However, training edge weights and kernel parameters with contrastive loss produces a learned distance metric with large margins between these graph categories. We demonstrate this by showing improved performance of a simple k-nearest-neighbors classifier on the learned distance matrix. We also demonstrate a further application of this method to biological image analysis: once trained, we use our model to compute the distance between the biological graphs and a set of graphs output by a cell division simulator. This allows us to identify simulation parameter regimes which are similar to each class of graph in our original dataset.

Yinan Wang, Diane Oyen, Weihong et al.

Catastrophic failure in brittle materials is often due to the rapid growth and coalescence of cracks aided by high internal stresses. Hence, accurate prediction of maximum internal stress is critical to predicting time to failure and improving the fracture resistance and reliability of materials. Existing high-fidelity methods, such as the Finite-Discrete Element Model (FDEM), are limited by their high computational cost. Therefore, to reduce computational cost while preserving accuracy, a novel deep learning model, "StressNet," is proposed to predict the entire sequence of maximum internal stress based on fracture propagation and the initial stress data. More specifically, the Temporal Independent Convolutional Neural Network (TI-CNN) is designed to capture the spatial features of fractures like fracture path and spall regions, and the Bidirectional Long Short-term Memory (Bi-LSTM) Network is adapted to capture the temporal features. By fusing these features, the evolution in time of the maximum internal stress can be accurately predicted. Moreover, an adaptive loss function is designed by dynamically integrating the Mean Squared Error (MSE) and the Mean Absolute Percentage Error (MAPE), to reflect the fluctuations in maximum internal stress. After training, the proposed model is able to compute accurate multi-step predictions of maximum internal stress in approximately 20 seconds, as compared to the FDEM run time of 4 hours, with an average MAPE of 2% relative to test data.

C. B. Scott, Eric Mjolsness

We define a novel type of ensemble Graph Convolutional Network (GCN) model. Using optimized linear projection operators to map between spatial scales of graph, this ensemble model learns to aggregate information from each scale for its final prediction. We calculate these linear projection operators as the infima of an objective function relating the structure matrices used for each GCN. Equipped with these projections, our model (a Graph Prolongation-Convolutional Network) outperforms other GCN ensemble models at predicting the potential energy of monomer subunits in a coarse-grained mechanochemical simulation of microtubule bending. We demonstrate these performance gains by measuring an estimate of the FLOPs spent to train each model, as well as wall-clock time. Because our model learns at multiple scales, it is possible to train at each scale according to a predetermined schedule of coarse vs. fine training. We examine several such schedules adapted from the Algebraic Multigrid (AMG) literature, and quantify the computational benefit of each. We also compare this model to another model which features an optimized coarsening of the input graph. Finally, we derive backpropagation rules for the input of our network model with respect to its output, and discuss how our method may be extended to very large graphs.

C. B. Scott, Eric Mjolsness

We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponential matrices, under norm-preserving and sparsity constraints. Variants of the distance metric are introduced to consider such optimized maps under sparsity constraints as well as fixed time-scaling between the two Laplacians. The objective function of this optimization is multimodal and has discontinuous slope, and is hence difficult for univariate optimizers to solve. We demonstrate a novel procedure for efficiently calculating these optima for two of our distance measure variants. We present numerical experiments demonstrating that (a) upper bounds of our distance metrics can be used to distinguish between lineages of related graphs; (b) our procedure is faster at finding the required optima, by as much as a factor of 10^3; and (c) the upper bounds satisfy the triangle inequality exactly under some assumptions and approximately under others. We also derive an upper bound for the distance between two graph products, in terms of the distance between the two pairs of factors. Additionally, we present several possible applications, including the construction of infinite "graph limits" by means of Cauchy sequences of graphs related to one another by our distance measure.

C. B. Scott, Eric Mjolsness

Multigrid modeling algorithms are a technique used to accelerate relaxation models running on a hierarchy of similar graphlike structures. We introduce and demonstrate a new method for training neural networks which uses multilevel methods. Using an objective function derived from a graph-distance metric, we perform orthogonally-constrained optimization to find optimal prolongation and restriction maps between graphs. We compare and contrast several methods for performing this numerical optimization, and additionally present some new theoretical results on upper bounds of this type of objective function. Once calculated, these optimal maps between graphs form the core of Multiscale Artificial Neural Network (MsANN) training, a new procedure we present which simultaneously trains a hierarchy of neural network models of varying spatial resolution. Parameter information is passed between members of this hierarchy according to standard coarsening and refinement schedules from the multiscale modelling literature. In our machine learning experiments, these models are able to learn faster than default training, achieving a comparable level of error in an order of magnitude fewer training examples.