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Researcher profile

Daniel Durstewitz

Daniel Durstewitz contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.DSphysics.comp-phArtificial IntelligenceDistributed, Parallel, and Cluster ComputingNeurons and Cognitionnlin.CDQuantitative Methods

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Published work

7 published item(s)

preprint2026arXiv

Parallel-in-Time Training of Recurrent Neural Networks for Dynamical Systems Reconstruction

Reconstructing nonlinear dynamical systems (DS) from data (DSR) is a fundamental challenge in science and engineering, but it inherently relies on sequential models. Recent breakthroughs for sequential models have produced algorithms that parallelize computation along sequence length $T$, achieving logarithmic time complexity, $\mathcal{O}(\log T)$. Since sequence lengths have been practically limited due to the linear runtime complexity $\mathcal{O}(T)$ of classical backpropagation through time, this opens new avenues for DSR. This paper studies two prominent classes of parallel-in-time algorithms for this task, both of which leverage parallel associative scans as their core computational primitive. The first class comprises models with linear yet non-autonomous dynamics and a nonlinear readout, such as modern State Space Models (SSMs), while the second consists of general nonlinear models which can be parallelized using the DEER framework. We find that the linear training-time recurrence of the first class of models imposes limitations that often hinder learning of accurate nonlinear dynamics. To address this, we augment DEER with Generalized Teacher Forcing (GTF), a novel variant within the more general nonlinear framework that ensures stable and effective learning of nonlinear dynamics across arbitrary sequence lengths. Using GTF-DEER, we investigate the benefits of training on extremely long sequences ($T>10^4$) for DSR. Our results show that access to such long trajectories significantly improves DSR if the data features long time scales. This work establishes GTF-DEER as a robust tool for data-driven discovery and underscores the largely untapped potential of long-sequence learning in modeling complex DS.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Reconstructing Nonlinear Dynamical Systems from Multi-Modal Time Series

Empirically observed time series in physics, biology, or medicine, are commonly generated by some underlying dynamical system (DS) which is the target of scientific interest. There is an increasing interest to harvest machine learning methods to reconstruct this latent DS in a data-driven, unsupervised way. In many areas of science it is common to sample time series observations from many data modalities simultaneously, e.g. electrophysiological and behavioral time series in a typical neuroscience experiment. However, current machine learning tools for reconstructing DSs usually focus on just one data modality. Here we propose a general framework for multi-modal data integration for the purpose of nonlinear DS reconstruction and the analysis of cross-modal relations. This framework is based on dynamically interpretable recurrent neural networks as general approximators of nonlinear DSs, coupled to sets of modality-specific decoder models from the class of generalized linear models. Both an expectation-maximization and a variational inference algorithm for model training are advanced and compared. We show on nonlinear DS benchmarks that our algorithms can efficiently compensate for too noisy or missing information in one data channel by exploiting other channels, and demonstrate on experimental neuroscience data how the algorithm learns to link different data domains to the underlying dynamics.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems

In many scientific disciplines, we are interested in inferring the nonlinear dynamical system underlying a set of observed time series, a challenging task in the face of chaotic behavior and noise. Previous deep learning approaches toward this goal often suffered from a lack of interpretability and tractability. In particular, the high-dimensional latent spaces often required for a faithful embedding, even when the underlying dynamics lives on a lower-dimensional manifold, can hamper theoretical analysis. Motivated by the emerging principles of dendritic computation, we augment a dynamically interpretable and mathematically tractable piecewise-linear (PL) recurrent neural network (RNN) by a linear spline basis expansion. We show that this approach retains all the theoretically appealing properties of the simple PLRNN, yet boosts its capacity for approximating arbitrary nonlinear dynamical systems in comparatively low dimensions. We employ two frameworks for training the system, one combining back-propagation-through-time (BPTT) with teacher forcing, and another based on fast and scalable variational inference. We show that the dendritically expanded PLRNN achieves better reconstructions with fewer parameters and dimensions on various dynamical systems benchmarks and compares favorably to other methods, while retaining a tractable and interpretable structure.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here we contribute to this topic by investigating the existence of n-cycles $(n\geq 3)$ and border-collision bifurcations in a class of n-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is shown that these n-cycles collide with the switching boundary in a border-collision bifurcation, and parametric regions for the existence of both stable and unstable n-cycles and border-collision bifurcations will be derived theoretically. We then discuss how our results can be extended and applied to PLRNNs. Finally, numerical simulations demonstrate the implementation of our results and are found to be in good agreement with the theoretical derivations. Our findings thus provide a basis for understanding periodic behavior in PLRNNs, how it emerges in bifurcations, and how it may lead into chaos.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Transformation of ReLU-based recurrent neural networks from discrete-time to continuous-time

Recurrent neural networks (RNN) as used in machine learning are commonly formulated in discrete time, i.e. as recursive maps. This brings a lot of advantages for training models on data, e.g. for the purpose of time series prediction or dynamical systems identification, as powerful and efficient inference algorithms exist for discrete time systems and numerical integration of differential equations is not necessary. On the other hand, mathematical analysis of dynamical systems inferred from data is often more convenient and enables additional insights if these are formulated in continuous time, i.e. as systems of ordinary (or partial) differential equations (ODE). Here we show how to perform such a translation from discrete to continuous time for a particular class of ReLU-based RNN. We prove three theorems on the mathematical equivalence between the discrete and continuous time formulations under a variety of conditions, and illustrate how to use our mathematical results on different machine learning and nonlinear dynamical systems examples.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our model is easily interpretable in neural terms, amenable to systematic dynamical systems analysis of the resulting set of equations, and can straightforwardly be transformed into an equivalent continuous-time dynamical system. The major contributions of this paper are the introduction of a new observation model suitable for functional magnetic resonance imaging (fMRI) coupled to the latent PLRNN, an efficient stepwise training procedure that forces the latent model to capture the 'true' underlying dynamics rather than just fitting (or predicting) the observations, and of an empirical measure based on the Kullback-Leibler divergence to evaluate from empirical time series how well this goal of approximating the underlying dynamics has been achieved. We validate and illustrate the power of our approach on simulated 'ground-truth' dynamical (benchmark) systems as well as on actual experimental fMRI time series, and demonstrate that the latent dynamics harbors task-related nonlinear structure that a linear dynamical model fails to capture. Given that fMRI is one of the most common techniques for measuring brain activity non-invasively in human subjects, this approach may provide a novel step toward analyzing aberrant (nonlinear) dynamics for clinical assessment or neuroscientific research.

0 citations0 reviewsNo code linkedOpen access