Deep learning for the solution of differential equations

Neuronové sítě se stávají čím dál tím populárnějším nástrojem pro řešení diferenciál- ních rovnic. Jejich použití ztělesňuje koncept physics-informed neural network (PINN), který kombinuje tradiční hlubokou neuronovou síť s fyzikálními zákony v podobě par- ciálních diferenciálních rovnic. Možnosti tohoto relativně nového přístupu prozkoumáme na třech rozmanitých příkladech, abychom mohli přehledně formulovat jeho výhody a nevýhody. Každý z problémů je také řešen metodou konečných prvků, která slouží jako referenční přístup. Kromě toho navrhujeme použití předtrénovaní, které se běžně používá v jiných vědeckých oborech. Pokud inicializujeme proces řešení rovnice pomocí výsledku podobného problému, významně tím zkrátíme výpočetní čas, který je zásadním nedostatkem PINN. 1Neural networks are becoming an ever more prominent method in the field of differen- tial equations. Their use is embodied in the concept of physics-informed neural network (PINN), which combines a traditional deep neural network with the underlying laws of physics described by PDEs. We investigate the abilities of this relatively novel approach on thee diverse examples in order to give a good overview of its advantages and issues. Every problem is also solved via the finite element method, which serves as a reference. In addition to that, we propose the usage of pre-training, which is already present in other scientific areas. If we initialize the process of solving of one equation with a solution to a similar problem, in some settings, we were able to significantly reduce computation time, which is major drawback of PINNs. 1Katedra numerické matematikyDepartment of Numerical MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Deep learning for the solution of differential equations

Neural networks are becoming an ever more prominent method in the field of differen- tial equations. Their use is embodied in the concept of physics-informed neural network (PINN), which combines a traditional deep neural network with the underlying laws of physics described by PDEs. We investigate the abilities of this relatively novel approach on thee diverse examples in order to give a good overview of its advantages and issues. Every problem is also solved via the finite element method, which serves as a reference. In addition to that, we propose the usage of pre-training, which is already present in other scientific areas. If we initialize the process of solving of one equation with a solution to a similar problem, in some settings, we were able to significantly reduce computation time, which is major drawback of PINNs.

Analytic and Holistic Cognitive Style as a Set of Independent Manifests: Evidence from a Validation Study of Six Measurement Instruments

Cognitive styles are commonly studied constructs in cognitive psychology. The theory of field dependence-independence was one of the most important cognitive styles. Yet in the past, its measurement had significant shortcomings in validity and reliability. The theory of analytic and holistic cognitive styles attempted to extend this theory and overcome its shortcomings. Unfortunately, the psychometric properties of its measurement methods were not properly verified. Furthermore, new statistical approaches, such as analysis of reaction times, have been overlooked by current research. The aim of this pre-registered study was to verify the psychometric properties (i.e., factor structure, split-half reliability, test-retest reliability, discriminant validity with intelligence and personality, and divergent, concurrent and predictive validity) of several methods routinely applied in the field. We developed/adapted six methods based on self-report questionnaires, rod-and-frame principles, embedded figures, and hierarchical figures. The analysis was conducted on 392 Czech participants, with two data collection waves. The results indicate that the use of methods based on the rod-and-frame principle may be unreliable, demonstrating no absence of association with intelligence. The use of embedded and hierarchical figures is recommended. The self-report questionnaire used in this study showed an unsatisfactory factor structure and also cannot be recommended without futher validation on independent samples. The findings also did not correspond with the original two-dimensional theory

Analytic and holistic cognitive style as a set of independent manifests: Evidence from a validation study of six measurement instruments.

Cognitive styles are commonly studied constructs in cognitive psychology. The theory of field dependence-independence was one of the most important cognitive styles. Yet in the past, its measurement had significant shortcomings in validity and reliability. The theory of analytic and holistic cognitive styles attempted to extend this theory and overcome its shortcomings. Unfortunately, the psychometric properties of its measurement methods were not properly verified. Furthermore, new statistical approaches, such as analysis of reaction times, have been overlooked by current research. The aim of this pre-registered study was to verify the psychometric properties (i.e., factor structure, split-half reliability, test-retest reliability, discriminant validity with intelligence and personality, and divergent, concurrent and predictive validity) of several methods routinely applied in the field. We developed/adapted six methods based on self-report questionnaires, rod-and-frame principles, embedded figures, and hierarchical figures. The analysis was conducted on 392 Czech participants, with two data collection waves. The results indicate that the use of methods based on the rod-and-frame principle may be unreliable, demonstrating no absence of association with intelligence. The use of embedded and hierarchical figures is recommended. The self-report questionnaire used in this study showed an unsatisfactory factor structure and also cannot be recommended without futher validation on independent samples. The findings also did not correspond with the original two-dimensional theory