Victor Boussange

Victor Boussange, Sebastian Becker, Arnulf Jentzen et al.

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman-Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

Victor Boussange, Bert Wuyts, Philipp Brun et al.

Biodiversity assessments are critically affected by the spatial scale at which species richness is measured. How species richness accumulates with sampling area depends on natural and anthropogenic processes whose effects can change depending on the spatial scale considered. These accumulation dynamics, described by the species-area relationship (SAR), are challenging to assess because most biodiversity surveys are restricted to sampling areas much smaller than the scales at which these processes operate. Here, we combine sampling theory and deep learning to predict local species richness within arbitrarily large sampling areas, enabling for the first time to estimate spatial differences in SARs. We demonstrate our approach by predicting vascular plant species richness across Europe and evaluate predictions against an independent dataset of plant community inventories. The resulting model, named deep SAR, delivers multi-scale species richness maps, improving coarse grain richness estimates by 32% compared to conventional methods, while delivering finer grain estimates. Additional to its predictive capabilities, we show how our deep SAR model can provide fundamental insights on the multi-scale effects of key biodiversity processes. The capacity of our approach to deliver comprehensive species richness estimates across the full spectrum of ecologically relevant scales is essential for robust biodiversity assessments and forecasts under global change.

Pau Vilimelis Aceituno, Jack William Miller, Noah Marti et al.

When training autoregressive models to forecast dynamical systems, a critical question arises: how far into the future should the model be trained to predict? Too short a horizon may miss long-term trends, while too long a horizon can impede convergence due to accumulating prediction errors. In this work, we formalize this trade-off by analyzing how the geometry of the loss landscape depends on the training horizon. We prove that for chaotic systems, the loss landscape's roughness grows exponentially with the training horizon, while for limit cycles, it grows linearly, making long-horizon training inherently challenging. However, we also show that models trained on long horizons generalize well to short-term forecasts, whereas those trained on short horizons suffer exponentially (resp. linearly) worse long-term predictions in chaotic (resp. periodic) systems. We validate our theory through numerical experiments and discuss practical implications for selecting training horizons. Our results provide a principled foundation for hyperparameter optimization in autoregressive forecasting models.