Machine Learning Topological Invariants with Neural Networks

In this Letter we supervisedly train neural networks to distinguish different topological phases in the context of topological band insulators. After training with Hamiltonians of one-dimensional insulators with chiral symmetry, the neural network can predict their topological winding numbers with nearly 100% accuracy, even for Hamiltonians with larger winding numbers that are not included in the training data. These results show a remarkable success that the neural network can capture the global and nonlinear topological features of quantum phases from local inputs. By opening up the neural network, we confirm that the network does learn the discrete version of the winding number formula. We also make a couple of remarks regarding the role of the symmetry and the opposite effect of regularization techniques when applying machine learning to physical systems.Comment: 6 pages, 4 figures and 1 table + 2 pages of supplemental materia

Additional file 1: Table S1. of Disorders of sex development: insights from targeted gene sequencing of a large international patient cohort

DSD gene variants. Each variant found in a diagnostic gene (after the filtering and curation process) is shown. In some cases where the gene is inherited in an autosomal recessive manner, two variants are grouped together. Inheritance has been indicated where familial samples were available: negative indicates negative for variant and N/A sample not available. De novo events have only been noted where both parental samples were available and found to be negative for the change. Previously reported refers to a variant being described in either ClinVar, HGMD, or a publication in a peer-reviewed journal via a PubMed search. Variants were classified consistent with previous MPS publications of DSD cohorts [8, 10] which were based on ACMG guidelines [15]. VUS were called for three reasons: 1 = fits phenotype but predicted to be benign; 2 = damaging but doesn’t fit phenotype; or 3 = variant in the AR repetitive region. Patients marked with an asterisk were identified to have two or more diagnostic gene variants. Null variants (frameshifts, splice sites mutations, and premature stop codons) are shown in bold. Patients have been classified based on clinical notes provided, according to the recommended classification of DSD in the Chicago consensus report. Classifications: CGD complete gonadal dysgenesis, DASA disorders of androgen synthesis or action, DSD DSD of “unknown” origin; hypospadias, LCH Leydig cell hypoplasia, OT ovotesticular DSD, PGD partial gonadal dysgenesis, PMDS persistent Müllerian duct syndrome; syndromic, T testicular DSD. Related affected individuals are indicated. File is in Excel spreadsheet format. (XLSX 47 kb