Economics-Inspired Neural Networks with Stabilizing Homotopies

Contemporary deep learning based solution methods used to compute approximate equilibria of high-dimensional dynamic stochastic economic models are often faced with two pain points. The first problem is that the loss function typically encodes a diverse set of equilibrium conditions, such as market clearing and households' or firms' optimality conditions. Hence the training algorithm trades off errors between those -- potentially very different -- equilibrium conditions. This renders the interpretation of the remaining errors challenging. The second problem is that portfolio choice in models with multiple assets is only pinned down for low errors in the corresponding equilibrium conditions. In the beginning of training, this can lead to fluctuating policies for different assets, which hampers the training process. To alleviate these issues, we propose two complementary innovations. First, we introduce Market Clearing Layers, a neural network architecture that automatically enforces all the market clearing conditions and borrowing constraints in the economy. Encoding economic constraints into the neural network architecture reduces the number of terms in the loss function and enhances the interpretability of the remaining equilibrium errors. Furthermore, we present a homotopy algorithm for solving portfolio choice problems with multiple assets, which ameliorates numerical instabilities arising in the context of deep learning. To illustrate our method we solve an overlapping generations model with two permanent risk aversion types, three distinct assets, and aggregate shocks

Products of small modules

summary:Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal

Classes of rings determined by a categorical property

Matematicko-fyzikální fakult

Gypsum endolithic phototrophs under moderate climate (Southern Sicily): their diversity and pigment composition

In this study, we used microscopic, spectroscopic, and molecular analysis to characterize endolithic colonization in gypsum (selenites and white crystalline gypsum) from several sites in Sicily. Our results showed that the dominant microorganisms in these environments are cyanobacteria, including: Chroococcidiopsis sp., Gloeocapsopsis pleurocapsoides, Gloeocapsa compacta, and Nostoc sp., as well as orange pigmented green microalgae from the Stephanospherinia clade. Single cell and filament sequencing coupled with 16S rRNA amplicon metagenomic profiling provided new insights into the phylogenetic and taxonomic diversity of the endolithic cyanobacteria. These organisms form differently pigmented zones within the gypsum. Our metagenomic profiling also showed differences in the taxonomic composition of endoliths in different gypsum varieties. Raman spectroscopy revealed that carotenoids were the most common pigments present in the samples. Other pigments such as gloeocapsin and scytonemin were also detected in the near-surface areas, suggesting that they play a significant role in the biology of endoliths in this environment. These pigments can be used as biomarkers for basic taxonomic identification, especially in case of cyanobacteria. The findings of this study provide new insights into the diversity and distribution of phototrophic microorganisms and their pigments in gypsum in Southern Sicily. Furthemore, this study highlights the complex nature of endolithic ecosystems and the effects of gypsum varieties on these communities, providing additional information on the general bioreceptivity of these environments.This project was supported by the Czech Science Foundation (Grant/Award No. 17-04270S and 21-03322S), Ministry of Education, Youth and Sports of the Czech Republic, National Programme of Sustainability I (Grant/Award No. LO1416), Charles University (Grant/Award Nos. UNCE/SCI/006 and UNCE 204069), ALGAMIC (Grant/Award No. CZ.1.05/2.1.00/19.0392). JM was supported by the Czech Science Foundation (GAČR) Project No. 22-06374S to accomplish phylogenetic and taxonomic analysis. JW was thankful for the financial support by the PGC2021-124362NB-I00 grant from MCI/AEI (Spain) and FEDER.Peer reviewe

Makro-epidemické modelování: Metoda hlubokého učení

V této práci prezentuji novou metodu pro výpočet globálně přesných řešení rekurzivních stochastických makro-epidemických modelů s potenciálně vysoko- dimenzionálním stavovým prostorem. V porovnání s existujícími studiemi, které buďto studují deterministické ekonomiky pomocí sekvenčních metod, nebo ana- lyzují stylizované modely řešitelné standardními metodami dynamického pro- gramování a lineární projekce, v této práci aplikuji algoritmus založený na hlubokém učení, který umožňuje analyzovat komplexní ekonomiky s agregátní nejistotou a velkým počtem stavových proměnných. Kromě řešení modelu vůči dané hodnotě parametrů prezentuji též rozšířený algoritmus, který umožňuje vyřešit celou množinu modelů indexovanou parametry reakční funkce vlády. Tento krok tak výrazně zjednodušuje výpočet optimální reakční funkce vlády, jelikož obchází nutnost opakovaného řešení modelu pro různé parametrizace vládní reakční funkce. 1I develop a novel method for computing globally accurate solutions to recur- sive macro-epidemic models featuring aggregate uncertainty and a potentially large number of state variables. Compared to the previous literature which either restricts attention to perfect-foresight economies amendable to sequence- space representation or focuses on highly simplified, low dimensional models that could can be analyzed using standard dynamic programming and linear projection techniques, I develop a deep learning-based algorithm that can han- dle rich environments featuring both aggregate uncertainty and large numbers of state variables. In addition to solving for particular model equilibria, I show how the deep learning method could be extended to solve for a whole set of models, indexed by the parameters of government policy. By pre-computing the whole equilibrium set, my deep learning method greatly simplifies compu- tation of optimal policies, since it bypasses the need to re-solve the model for many different values of policy parameters. 1CERGEFakulta sociálních vědFaculty of Social Science