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Daniel Durstewitz

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15 papers

2 author rows

ICLR Conference 2025 Conference Paper

Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data

Manuel Brenner
Elias Weber
Georgia Koppe
Daniel Durstewitz

In science, we are often interested in obtaining a generative model of the underlying system dynamics from observed time series. While powerful methods for dynamical systems reconstruction (DSR) exist when data come from a single domain, how to best integrate data from multiple dynamical regimes and leverage it for generalization is still an open question. This becomes particularly important when individual time series are short, and group-level information may help to fill in for gaps in single-domain data. Here we introduce a hierarchical framework that enables to harvest group-level (multi-domain) information while retaining all single-domain characteristics, and showcase it on popular DSR benchmarks, as well as on neuroscience and medical data. In addition to faithful reconstruction of all individual dynamical regimes, our unsupervised methodology discovers common low-dimensional feature spaces in which datasets with similar dynamics cluster. The features spanning these spaces were further dynamically highly interpretable, surprisingly in often linear relation to control parameters that govern the dynamics of the underlying system. Finally, we illustrate transfer learning and generalization to new parameter regimes, paving the way toward DSR foundation models.

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NeurIPS Conference 2025 Conference Paper

True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics

Christoph Jürgen Hemmer
Daniel Durstewitz

Complex, temporally evolving phenomena, from climate to brain activity, are governed by dynamical systems (DS). DS reconstruction (DSR) seeks to infer generative surrogate models of these from observed data, reproducing their long-term behavior. Existing DSR approaches require purpose-training for any new system observed, lacking the zero-shot and in-context inference capabilities known from LLMs. Here we introduce DynaMix, a novel multivariate ALRNN-based mixture-of-experts architecture pre-trained for DSR, the first DSR model able to generalize zero-shot to out-of-domain DS. Just from a provided context signal, without any re-training, DynaMix faithfully forecasts the long-term evolution of novel DS where existing time series (TS) foundation models, like Chronos, fail -- at a fraction of the number of parameters (0. 1%) and orders of magnitude faster inference times. DynaMix outperforms TS foundation models in terms of long-term statistics, and often also short-term forecasts, even on real-world time series, like traffic or weather data, typically used for training and evaluating TS models, but not at all part of DynaMix' training corpus. We illustrate some of the failure modes of TS models for DSR problems, and conclude that models built on DS principles may bear a huge potential also for advancing the TS prediction field.

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NeurIPS Conference 2024 Conference Paper

A scalable generative model for dynamical system reconstruction from neuroimaging data

Eric Volkmann
Alena Brändle
Daniel Durstewitz
Georgia Koppe

Data-driven inference of the generative dynamics underlying a set of observed time series is of growing interest in machine learning and the natural sciences. In neuroscience, such methods promise to alleviate the need to handcraft models based on biophysical principles and allow to automatize the inference of inter-individual differences in brain dynamics. Recent breakthroughs in training techniques for state space models (SSMs) specifically geared toward dynamical systems (DS) reconstruction (DSR) enable to recover the underlying system including its geometrical (attractor) and long-term statistical invariants from even short time series. These techniques are based on control-theoretic ideas, like modern variants of teacher forcing (TF), to ensure stable loss gradient propagation while training. However, as it currently stands, these techniques are not directly applicable to data modalities where current observations depend on an entire history of previous states due to a signal’s filtering properties, as common in neuroscience (and physiology more generally). Prominent examples are the blood oxygenation level dependent (BOLD) signal in functional magnetic resonance imaging (fMRI) or Ca$^{2+}$ imaging data. Such types of signals render the SSM's decoder model non-invertible, a requirement for previous TF-based methods. Here, exploiting the recent success of control techniques for training SSMs, we propose a novel algorithm that solves this problem and scales exceptionally well with model dimensionality and filter length. We demonstrate its efficiency in reconstructing dynamical systems, including their state space geometry and long-term temporal properties, from just short BOLD time series.

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NeurIPS Conference 2024 Conference Paper

Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction

Manuel Brenner
Christoph Jürgen Hemmer
Zahra Monfared
Daniel Durstewitz

Dynamical systems theory (DST) is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS combined by switching manifolds, i. e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and Rössler systems, AL-RNNs derive, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

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ICML Conference 2024 Conference Paper

Integrating Multimodal Data for Joint Generative Modeling of Complex Dynamics

Manuel Brenner
Florian Hess
Georgia Koppe
Daniel Durstewitz

Many, if not most, systems of interest in science are naturally described as nonlinear dynamical systems. Empirically, we commonly access these systems through time series measurements. Often such time series may consist of discrete random variables rather than continuous measurements, or may be composed of measurements from multiple data modalities observed simultaneously. For instance, in neuroscience we may have behavioral labels in addition to spike counts and continuous physiological recordings. While by now there is a burgeoning literature on deep learning for dynamical systems reconstruction (DSR), multimodal data integration has hardly been considered in this context. Here we provide such an efficient and flexible algorithmic framework that rests on a multimodal variational autoencoder for generating a sparse teacher signal that guides training of a reconstruction model, exploiting recent advances in DSR training techniques. It enables to combine various sources of information for optimal reconstruction, even allows for reconstruction from symbolic data (class labels) alone, and connects different types of observations within a common latent dynamics space. In contrast to previous multimodal data integration techniques for scientific applications, our framework is fully generative, producing, after training, trajectories with the same geometrical and temporal structure as those of the ground truth system.

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ICML Conference 2024 Conference Paper

Optimal Recurrent Network Topologies for Dynamical Systems Reconstruction

Christoph Jürgen Hemmer
Manuel Brenner
Florian Hess
Daniel Durstewitz

In dynamical systems reconstruction (DSR) we seek to infer from time series measurements a generative model of the underlying dynamical process. This is a prime objective in any scientific discipline, where we are particularly interested in parsimonious models with a low parameter load. A common strategy here is parameter pruning, removing all parameters with small weights. However, here we find this strategy does not work for DSR, where even low magnitude parameters can contribute considerably to the system dynamics. On the other hand, it is well known that many natural systems which generate complex dynamics, like the brain or ecological networks, have a sparse topology with comparatively few links. Inspired by this, we show that geometric pruning, where in contrast to magnitude-based pruning weights with a low contribution to an attractor’s geometrical structure are removed, indeed manages to reduce parameter load substantially without significantly hampering DSR quality. We further find that the networks resulting from geometric pruning have a specific type of topology, and that this topology, and not the magnitude of weights, is what is most crucial to performance. We provide an algorithm that automatically generates such topologies which can be used as priors for generative modeling of dynamical systems by RNNs, and compare it to other well studied topologies like small-world or scale-free networks.

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ICML Conference 2024 Conference Paper

Out-of-Domain Generalization in Dynamical Systems Reconstruction

Niclas Alexander Göring
Florian Hess
Manuel Brenner
Zahra Monfared
Daniel Durstewitz

In science we are interested in finding the governing equations, the dynamical rules, underlying empirical phenomena. While traditionally scientific models are derived through cycles of human insight and experimentation, recently deep learning (DL) techniques have been advanced to reconstruct dynamical systems (DS) directly from time series data. State-of-the-art dynamical systems reconstruction (DSR) methods show promise in capturing invariant and long-term properties of observed DS, but their ability to generalize to unobserved domains remains an open challenge. Yet, this is a crucial property we would expect from any viable scientific theory. In this work, we provide a formal framework that addresses generalization in DSR. We explain why and how out-of-domain (OOD) generalization (OODG) in DSR profoundly differs from OODG considered elsewhere in machine learning. We introduce mathematical notions based on topological concepts and ergodic theory to formalize the idea of learnability of a DSR model. We formally prove that black-box DL techniques, without adequate structural priors, generally will not be able to learn a generalizing DSR model. We also show this empirically, considering major classes of DSR algorithms proposed so far, and illustrate where and why they fail to generalize across the whole phase space. Our study provides the first comprehensive mathematical treatment of OODG in DSR, and gives a deeper conceptual understanding of where the fundamental problems in OODG lie and how they could possibly be addressed in practice.

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NeurIPS Conference 2023 Conference Paper

Bifurcations and loss jumps in RNN training

Lukas Eisenmann
Zahra Monfared
Niclas Göring
Daniel Durstewitz

Recurrent neural networks (RNNs) are popular machine learning tools for modeling and forecasting sequential data and for inferring dynamical systems (DS) from observed time series. Concepts from DS theory (DST) have variously been used to further our understanding of both, how trained RNNs solve complex tasks, and the training process itself. Bifurcations are particularly important phenomena in DS, including RNNs, that refer to topological (qualitative) changes in a system's dynamical behavior as one or more of its parameters are varied. Knowing the bifurcation structure of an RNN will thus allow to deduce many of its computational and dynamical properties, like its sensitivity to parameter variations or its behavior during training. In particular, bifurcations may account for sudden loss jumps observed in RNN training that could severely impede the training process. Here we first mathematically prove for a particular class of ReLU-based RNNs that certain bifurcations are indeed associated with loss gradients tending toward infinity or zero. We then introduce a novel heuristic algorithm for detecting all fixed points and $k$-cycles in ReLU-based RNNs and their existence and stability regions, hence bifurcation manifolds in parameter space. In contrast to previous numerical algorithms for finding fixed points and common continuation methods, our algorithm provides $\textit{exact}$ results and returns fixed points and cycles up to high orders with surprisingly good scaling behavior. We exemplify the algorithm on the analysis of the training process of RNNs, and find that the recently introduced technique of generalized teacher forcing completely avoids certain types of bifurcations in training. Thus, besides facilitating the DST analysis of trained RNNs, our algorithm provides a powerful instrument for analyzing the training process itself.

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ICML Conference 2023 Conference Paper

Generalized Teacher Forcing for Learning Chaotic Dynamics

Florian Hess
Zahra Monfared
Manuel Brenner
Daniel Durstewitz

Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we are interested in reconstructing such systems from observed time series for prediction or mechanistic insight, where by reconstruction we mean learning geometrical and invariant temporal properties of the system in question (like attractors). However, training reconstruction algorithms like recurrent neural networks (RNNs) on such systems by gradient-descent based techniques faces severe challenges. This is mainly due to exploding gradients caused by the exponential divergence of trajectories in chaotic systems. Moreover, for (scientific) interpretability we wish to have as low dimensional reconstructions as possible, preferably in a model which is mathematically tractable. Here we report that a surprisingly simple modification of teacher forcing leads to provably strictly all-time bounded gradients in training on chaotic systems, and, when paired with a simple architectural rearrangement of a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful reconstruction in spaces of at most the dimensionality of the observed system. We show on several DS that with these amendments we can reconstruct DS better than current SOTA algorithms, in much lower dimensions. Performance differences were particularly compelling on real world data with which most other methods severely struggled. This work thus led to a simple yet powerful DS reconstruction algorithm which is highly interpretable at the same time.

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NeurIPS Conference 2022 Conference Paper

On the difficulty of learning chaotic dynamics with RNNs

Jonas Mikhaeil
Zahra Monfared
Daniel Durstewitz

Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or - more recently - imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, however, convey severe limitations on the expressivity of the RNN. Essential intrinsic dynamics such as multistability or chaos are disabled. This is inherently at disaccord with the chaotic nature of many, if not most, time series encountered in nature and society. It is particularly problematic in scientific applications where one aims to reconstruct the underlying dynamical system. Here we offer a comprehensive theoretical treatment of this problem by relating the loss gradients during RNN training to the Lyapunov spectrum of RNN-generated orbits. We mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNNs with chaotic dynamics always diverge. Based on these analyses and insights we suggest ways of how to optimize the training process on chaotic data according to the system's Lyapunov spectrum, regardless of the employed RNN architecture.

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ICML Conference 2022 Conference Paper

Reconstructing Nonlinear Dynamical Systems from Multi-Modal Time Series

Daniel Kramer
Philine Lou Bommer
Daniel Durstewitz
Carlo Tombolini
Georgia Koppe

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.

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ICML Conference 2022 Conference Paper

Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems

Manuel Brenner
Florian Hess
Jonas M. Mikhaeil
Leonard F. Bereska
Zahra Monfared
Po-Chen Kuo
Daniel Durstewitz

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 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.

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ICLR Conference 2021 Conference Paper

Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies

Dominik Schmidt
Georgia Koppe
Zahra Monfared
Max Beutelspacher
Daniel Durstewitz

A main theoretical interest in biology and physics is to identify the nonlinear dynamical system (DS) that generated observed time series. Recurrent Neural Networks (RNN) are, in principle, powerful enough to approximate any underlying DS, but in their vanilla form suffer from the exploding vs. vanishing gradients problem. Previous attempts to alleviate this problem resulted either in more complicated, mathematically less tractable RNN architectures, or strongly limited the dynamical expressiveness of the RNN. Here we address this issue by suggesting a simple regularization scheme for vanilla RNN with ReLU activation which enables them to solve long-range dependency problems and express slow time scales, while retaining a simple mathematical structure which makes their DS properties partly analytically accessible. We prove two theorems that establish a tight connection between the regularized RNN dynamics and their gradients, illustrate on DS benchmarks that our regularization approach strongly eases the reconstruction of DS which harbor widely differing time scales, and show that our method is also en par with other long-range architectures like LSTMs on several tasks.

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ICML Conference 2020 Conference Paper

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

Zahra Monfared
Daniel Durstewitz

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/ PDE). 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.

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NeurIPS Conference 2017 Conference Paper

Sparse convolutional coding for neuronal assembly detection

Sven Peter
Elke Kirschbaum
Martin Both
Lee Campbell
Brandon Harvey
Conor Heins
Daniel Durstewitz
Ferran Diego

Cell assemblies, originally proposed by Donald Hebb (1949), are subsets of neurons firing in a temporally coordinated way that gives rise to repeated motifs supposed to underly neural representations and information processing. Although Hebb's original proposal dates back many decades, the detection of assemblies and their role in coding is still an open and current research topic, partly because simultaneous recordings from large populations of neurons became feasible only relatively recently. Most current and easy-to-apply computational techniques focus on the identification of strictly synchronously spiking neurons. In this paper we propose a new algorithm, based on sparse convolutional coding, for detecting recurrent motifs of arbitrary structure up to a given length. Testing of our algorithm on synthetically generated datasets shows that it outperforms established methods and accurately identifies the temporal structure of embedded assemblies, even when these contain overlapping neurons or when strong background noise is present. Moreover, exploratory analysis of experimental datasets from hippocampal slices and cortical neuron cultures have provided promising results.

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