TY - JOUR A1 - Hartmann, Carsten A1 - Richter, Lorenz A1 - Schütte, Christof A1 - Zhang, Wei T1 - Variational characterization of free energy: theory and algorithms JF - Entropy Y1 - 2017 U6 - https://doi.org/10.3390/e19110626 VL - 19 IS - 11 ER - TY - JOUR A1 - Richter, Lorenz A1 - Sallandt, Leon A1 - Nüsken, Nikolas T1 - From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs JF - Journal of Machine Learning Research N2 - The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the romise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects. Y1 - 2024 UR - https://www.jmlr.org/papers/volume25/23-0982/23-0982.pdf VL - 25 SP - 248 ER - TY - JOUR A1 - Sun, Jingtong A1 - Berner, Julius A1 - Richter, Lorenz A1 - Zeinhofer, Marius A1 - Müller, Johannes A1 - Azizzadenesheli, Kamyar A1 - Anandkumar, Anima T1 - Dynamical Measure Transport and Neural PDE Solvers for Sampling N2 - The task of sampling from a probability density can be approached as transporting a tractable density function to the target, known as dynamical measure transport. In this work, we tackle it through a principled unified framework using deterministic or stochastic evolutions described by partial differential equations (PDEs). This framework incorporates prior trajectory-based sampling methods, such as diffusion models or Schrödinger bridges, without relying on the concept of time-reversals. Moreover, it allows us to propose novel numerical methods for solving the transport task and thus sampling from complicated targets without the need for the normalization constant or data samples. We employ physics-informed neural networks (PINNs) to approximate the respective PDE solutions, implying both conceptional and computational advantages. In particular, PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently, leading to significantly better mode coverage in the sampling task compared to alternative methods. Moreover, they can readily be fine-tuned with Gauss-Newton methods to achieve high accuracy in sampling. Y1 - 2024 ER - TY - JOUR A1 - Weilandt, Frank A1 - Behling, Robert A1 - Goncalves, Romulo A1 - Madadi, Arash A1 - Richter, Lorenz A1 - Sanona, Tiago A1 - Spengler, Daniel A1 - Welsch, Jona T1 - Early Crop Classification via Multi-Modal Satellite Data Fusion and Temporal Attention JF - Remote Sensing N2 - In this article, we propose a deep learning-based algorithm for the classification of crop types from Sentinel-1 and Sentinel-2 time series data which is based on the celebrated transformer architecture. Crucially, we enable our algorithm to do early classification, i.e., predict crop types at arbitrary time points early in the year with a single trained model (progressive intra-season classification). Such early season predictions are of practical relevance for instance for yield forecasts or the modeling of agricultural water balances, therefore being important for the public as well as the private sector. Furthermore, we improve the mechanism of combining different data sources for the prediction task, allowing for both optical and radar data as inputs (multi-modal data fusion) without the need for temporal interpolation. We can demonstrate the effectiveness of our approach on an extensive data set from three federal states of Germany reaching an average F1 score of 0.92 using data of a complete growing season to predict the eight most important crop types and an F1 score above 0.8 when doing early classification at least one month before harvest time. In carefully chosen experiments, we can show that our model generalizes well in time and space. Y1 - 2023 U6 - https://doi.org/10.3390/rs15030799 VL - 15 IS - 3 SP - 799 ER - TY - JOUR A1 - Reuss, Joana A1 - Macdonald, Jan A1 - Becker, Simon A1 - Richter, Lorenz A1 - Körner, Marco T1 - The EuroCropsML time series benchmark dataset for few-shot crop type classification in Europe JF - Scientific Data N2 - We introduce EuroCropsML, an analysis-ready remote sensing dataset based on the open-source EuroCrops collection, for machine learning (ML) benchmarking of time series crop type classification in Europe. It is the first time-resolved remote sensing dataset designed to benchmark transnational few-shot crop type classification algorithms that supports advancements in algorithmic development and research comparability. It comprises 706683 multi-class labeled data points across 176 crop classes. Each data point features a time series of per-parcel median pixel values extracted from Sentinel-2 L1C data and precise geospatial coordinates. EuroCropsML is publicly available on Zenodo. Y1 - 2025 U6 - https://doi.org/10.1038/s41597-025-04952-7 VL - 12 ER - TY - JOUR A1 - Blessing, Denis A1 - Berner, Julius A1 - Richter, Lorenz A1 - Domingo-Enrich, Carles A1 - Du, Yuanqi A1 - Vahdat, Arash A1 - Neumann, Gerhard T1 - Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference N2 - Solving stochastic optimal control problems with quadratic control costs can be viewed as approximating a target path space measure, e.g. via gradient-based optimization. In practice, however, this optimization is challenging in particular if the target measure differs substantially from the prior. In this work, we therefore approach the problem by iteratively solving constrained problems incorporating trust regions that aim for approaching the target measure gradually in a systematic way. It turns out that this trust region based strategy can be understood as a geometric annealing from the prior to the target measure, where, however, the incorporated trust regions lead to a principled and educated way of choosing the time steps in the annealing path. We demonstrate in multiple optimal control applications that our novel method can improve performance significantly, including tasks in diffusion-based sampling, transition path sampling, and fine-tuning of diffusion models. Y1 - 2025 ER - TY - JOUR A1 - Berner, Julius A1 - Richter, Lorenz A1 - Sendera, Marcin A1 - Rector-Brooks, Jarrid A1 - Malkin, Nikolay T1 - From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training N2 - We study the problem of training neural stochastic differential equations, or diffusion models, to sample from a Boltzmann distribution without access to target samples. Existing methods for training such models enforce time-reversal of the generative and noising processes, using either differentiable simulation or off-policy reinforcement learning (RL). We prove equivalences between families of objectives in the limit of infinitesimal discretization steps, linking entropic RL methods (GFlowNets) with continuous-time objects (partial differential equations and path space measures). We further show that an appropriate choice of coarse time discretization during training allows greatly improved sample efficiency and the use of time-local objectives, achieving competitive performance on standard sampling benchmarks with reduced computational cost. Y1 - 2025 ER - TY - JOUR A1 - Reuss, Joana A1 - Macdonald, Jan A1 - Becker, Simon A1 - Schultka, Konrad A1 - Richter, Lorenz A1 - Körner, Marco T1 - Meta-learning For Few-Shot Time Series Crop Type Classification: A Benchmark On The EuroCropsML Dataset N2 - Spatial imbalances in crop type data pose significant challenges for accurate classification in remote sensing applications. Algorithms aiming at transferring knowledge from data-rich to data-scarce tasks have thus surged in popularity. However, despite their effectiveness in previous evaluations, their performance in challenging real-world applications is unclear and needs to be evaluated. This study benchmarks transfer learning and several meta-learning algorithms, including (First-Order) Model-Agnostic Meta-Learning ((FO)-MAML), Almost No Inner Loop (ANIL), and Task-Informed Meta-Learning (TIML), on the real-world EuroCropsML time series dataset, which combines farmer-reported crop data with Sentinel-2 satellite observations from Estonia, Latvia, and Portugal. Our findings indicate that MAML-based meta-learning algorithms achieve slightly higher accuracy compared to simpler transfer learning methods when applied to crop type classification tasks in Estonia after pre-training on data from Latvia. However, this improvement comes at the cost of increased computational demands and training time. Moreover, we find that the transfer of knowledge between geographically disparate regions, such as Estonia and Portugal, poses significant challenges to all investigated algorithms. These insights underscore the trade-offs between accuracy and computational resource requirements in selecting machine learning methods for real-world crop type classification tasks and highlight the difficulties of transferring knowledge between different regions of the Earth. To facilitate future research in this domain, we present the first comprehensive benchmark for evaluating transfer and meta-learning methods for crop type classification under real-world conditions. The corresponding code is publicly available at this https URL. Y1 - 2025 ER - TY - JOUR A1 - Ribera Borrell, Enric A1 - Quer, Jannes A1 - Richter, Lorenz A1 - Schütte, Christof T1 - Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics JF - SIAM Journal on Scientific Computing (SISC) N2 - Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings. KW - importance sampling KW - stochastic optimal control KW - rare event simulation KW - metastability KW - neural networks KW - metadynamics Y1 - 2023 U6 - https://doi.org/10.1137/22M1503464 VL - 89 IS - 1 ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Richter, Lorenz T1 - Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems JF - Journal of Machine Learning N2 - Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of backward stochastic differential equations (BSDEs) and those aiming to minimize a regression-type L2-error (physics-informed neural networks, PINNs). In this paper, we review the literature and suggest a methodology based on the novel diffusion loss that interpolates between BSDEs and PINNs. Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BSDEs and PINNs. The diffusion loss furthermore bears close similarities to (least squares) temporal difference objectives found in reinforcement learning. We also discuss eigenvalue problems and perform extensive numerical studies, including calculations of the ground state for nonlinear Schr ¨odinger operators and committor functions relevant in molecular dynamics. Y1 - 2023 U6 - https://doi.org/0.4208/jml.220416 VL - 2 IS - 1 SP - 31 EP - 64 ER -