A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients

In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem to possess the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the dimension $d \in \mathbb{N}$ of the function which the DNN aims to approximate in such computational problems. There is also a large number of rigorous mathematical approximation results for artificial neural networks in the scientific literature but there are only a few special situations where results in the literature can rigorously justify the success of DNNs in high-dimensional function approximation. The key contribution of this paper is to reveal that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the PDE dimension $d \in \mathbb{N}$ and the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$. A crucial ingredient in our proof is the fact that the artificial neural network used to approximate the solution of the PDE is indeed a deep artificial neural network with a large number of hidden layers.Comment: 48 page

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are $n$-times continuously Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also $n$-times continuously Fr\'{e}chet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes

Methane production and oxidation potentials along a fen-bog gradient from southern boreal to subarctic peatlands in Finland

Methane (CH4) emissions from northern peatlands are projected to increase due to climate change, primarily because of projected increases in soil temperature. Yet, the rates and temperature responses of the two CH4 emission-related microbial processes (CH4 production by methanogens and oxidation by methanotrophs) are poorly known. Further, peatland sites within a fen-bog gradient are known to differ in the variables that regulate these two mechanisms, yet the interaction between peatland type and temperature lacks quantitative understanding. Here, we investigated potential CH4 production and oxidation rates for 14 peatlands in Finland located between c. 60 and 70 degrees N latitude, representing bogs, poor fens, and rich fens. Potentials were measured at three different temperatures (5, 17.5, and 30celcius) using the laboratory incubation method. We linked CH4 production and oxidation patterns to their methanogen and methanotroph abundance, peat properties, and plant functional types. We found that the rich fen-bog gradient-related nutrient availability and methanogen abundance increased the temperature response of CH4 production, with rich fens exhibiting the greatest production potentials. Oxidation potential showed a steeper temperature response than production, which was explained by aerenchymous plant cover, peat water holding capacity, peat nitrogen, and sulfate content. The steeper temperature response of oxidation suggests that, at higher temperatures, CH4 oxidation might balance increased CH4 production. Predicting net CH4 fluxes as an outcome of the two mechanisms is complicated due to their different controls and temperature responses. The lack of correlation between field CH4 fluxes and production/oxidation potentials, and the positive correlation with aerenchymous plants points toward the essential role of CH4 transport for emissions. The scenario of drying peatlands under climate change, which is likely to promote Sphagnum establishment over brown mosses in many places, will potentially reduce the predicted warming-related increase in CH4 emissions by shifting rich fens to Sphagnum-dominated systems.Peer reviewe

Multi-decadal improvements in the ecological quality of European rivers are not consistently reflected in biodiversity metrics

Humans impact terrestrial, marine and freshwater ecosystems, yet many broad-scale studies have found no systematic, negative biodiversity changes (for example, decreasing abundance or taxon richness). Here we show that mixed biodiversity responses may arise because community metrics show variable responses to anthropogenic impacts across broad spatial scales. We first quantified temporal trends in anthropogenic impacts for 1,365 riverine invertebrate communities from 23 European countries, based on similarity to least-impacted reference communities. Reference comparisons provide necessary, but often missing, baselines for evaluating whether communities are negatively impacted or have improved (less or more similar, respectively). We then determined whether changing impacts were consistently reflected in metrics of community abundance, taxon richness, evenness and composition. Invertebrate communities improved, that is, became more similar to reference conditions, from 1992 until the 2010s, after which improvements plateaued. Improvements were generally reflected by higher taxon richness, providing evidence that certain community metrics can broadly indicate anthropogenic impacts. However, richness responses were highly variable among sites, and we found no consistent responses in community abundance, evenness or composition. These findings suggest that, without sufficient data and careful metric selection, many common community metrics cannot reliably reflect anthropogenic impacts, helping explain the prevalence of mixed biodiversity trends.We thank J. England for assistance with calculating ecological quality and the biomonitoring indices in the UK. Funding for authors, data collection and processing was provided by the European Union Horizon 2020 project eLTER PLUS (grant number 871128). F.A. was supported by the Swiss National Science Foundation (grant numbers 310030_197410 and 31003A_173074) and the University of Zurich Research Priority Program Global Change and Biodiversity. J.B. and M.A.-C. were funded by the European Commission, under the L‘Instrument Financier pour l’Environnement (LIFE) Nature and Biodiversity program, as part of the project LIFE-DIVAQUA (LIFE18 NAT/ES/000121) and also by the project ‘WATERLANDS’ (PID2019-107085RB-I00) funded by the Ministerio de Ciencia, Innovación y Universidades (MCIN) and Agencia Estatal de Investigación (AEI; MCIN/AEI/10.13039/501100011033/ and by the European Regional Development Fund (ERDF) ‘A way of making Europe’. N.J.B. and V.P. were supported by the Lithuanian Environmental Protection Agency (https://aaa.lrv.lt/) who collected the data and were funded by the Lithuanian Research Council (project number S-PD-22-72). J.H. was supported by the Academy of Finland (grant number 331957). S.C.J. acknowledges funding by the Leibniz Competition project Freshwater Megafauna Futures and the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung or BMBF; 033W034A). A.L. acknowledges funding by the Spanish Ministry of Science and Innovation (PID2020-115830GB-100). P.P., M.P. and M.S. were supported by the Czech Science Foundation (GA23-05268S and P505-20-17305S) and thank the Czech Hydrometeorological Institute and the state enterprises Povodí for the data used to calculate ecological quality metrics from the Czech surface water monitoring program. H.T. was supported by the Estonian Research Council (number PRG1266) and by the Estonian national program ‘Humanitarian and natural science collections’. M.J.F. acknowledges the support of Fundação para a Ciência e Tecnologia, Portugal, through the projects UIDB/04292/2020 and UIDP/04292/2020 granted to the Marine and Environmental Sciences Centre, LA/P/0069/2020 granted to the Associate Laboratory Aquatic Research Network (ARNET), and a Call Estímulo ao Emprego Científico (CEEC) contract.Peer reviewe

The recovery of European freshwater biodiversity has come to a halt

Owing to a long history of anthropogenic pressures, freshwater ecosystems are among the most vulnerable to biodiversity loss1. Mitigation measures, including wastewater treatment and hydromorphological restoration, have aimed to improve environmental quality and foster the recovery of freshwater biodiversity2. Here, using 1,816 time series of freshwater invertebrate communities collected across 22 European countries between 1968 and 2020, we quantified temporal trends in taxonomic and functional diversity and their responses to environmental pressures and gradients. We observed overall increases in taxon richness (0.73% per year), functional richness (2.4% per year) and abundance (1.17% per year). However, these increases primarily occurred before the 2010s, and have since plateaued. Freshwater communities downstream of dams, urban areas and cropland were less likely to experience recovery. Communities at sites with faster rates of warming had fewer gains in taxon richness, functional richness and abundance. Although biodiversity gains in the 1990s and 2000s probably reflect the effectiveness of water-quality improvements and restoration projects, the decelerating trajectory in the 2010s suggests that the current measures offer diminishing returns. Given new and persistent pressures on freshwater ecosystems, including emerging pollutants, climate change and the spread of invasive species, we call for additional mitigation to revive the recovery of freshwater biodiversity.N. Kaffenberger helped with initial data compilation. Funding for authors and data collection and processing was provided by the EU Horizon 2020 project eLTER PLUS (grant agreement no. 871128); the German Federal Ministry of Education and Research (BMBF; 033W034A); the German Research Foundation (DFG FZT 118, 202548816); Czech Republic project no. P505-20-17305S; the Leibniz Competition (J45/2018, P74/2018); the Spanish Ministerio de Economía, Industria y Competitividad—Agencia Estatal de Investigación and the European Regional Development Fund (MECODISPER project CTM 2017-89295-P); Ramón y Cajal contracts and the project funded by the Spanish Ministry of Science and Innovation (RYC2019-027446-I, RYC2020-029829-I, PID2020-115830GB-100); the Danish Environment Agency; the Norwegian Environment Agency; SOMINCOR—Lundin mining & FCT—Fundação para a Ciência e Tecnologia, Portugal; the Swedish University of Agricultural Sciences; the Swiss National Science Foundation (grant PP00P3_179089); the EU LIFE programme (DIVAQUA project, LIFE18 NAT/ES/000121); the UK Natural Environment Research Council (GLiTRS project NE/V006886/1 and NE/R016429/1 as part of the UK-SCAPE programme); the Autonomous Province of Bolzano (Italy); and the Estonian Research Council (grant no. PRG1266), Estonian National Program ‘Humanitarian and natural science collections’. The Environment Agency of England, the Scottish Environmental Protection Agency and Natural Resources Wales provided publicly available data. We acknowledge the members of the Flanders Environment Agency for providing data. This article is a contribution of the Alliance for Freshwater Life (www.allianceforfreshwaterlife.org).Peer reviewe