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Title: Inferring Dynamical Systems with Long-Range Dependencies through Line Attractor Regularization

Paper content:
Vanilla RNN with ReLU activation have a simple structure that is amenable to systematic dynamical systems analysis and interpretation, but they suffer from the exploding vs. vanishing gradients problem.
Recent attempts to retain this simplicity while alleviating the gradient problem are based on proper initialization schemes or orthogonality/unitary constraints on the RNN’s recurrency matrix, which, however, comes with limitations to its expressive power with regards to dynamical systems phenomena like chaos or multi-stability.
Here, we instead suggest a regularization scheme that pushes part of the RNN’s latent subspace toward a line attractor configuration that enables long short-term memory and arbitrarily slow time scales.
We show that our approach excels on a number of benchmarks like the sequential MNIST or multiplication problems, and enables reconstruction of dynamical systems which harbor widely different time scales.
Theories of complex systems in biology and physics are often formulated in terms of sets of stochastic differential or difference equations, i.e. as stochastic dynamical systems (DS).
A long-standing desire is to retrieve these generating dynamical equations directly from observed time series data (Kantz & Schreiber, 2004) .
A variety of machine and deep learning methodologies toward this goal have been introduced in recent years (Chen et al., 2017; Champion et al., 2019; Jordan et al., 2019; Duncker et al., 2019; Ayed et al., 2019; Durstewitz, 2017; Koppe et al., 2019) , many of them based on recurrent neural networks (RNN) which can universally approximate any DS (i.e., its flow field) under some mild conditions (Funahashi & Nakamura, 1993; Kimura & Nakano, 1998) .
However, vanilla RNN as often used in this context are well known for their problems in capturing long-term dependencies and slow time scales in the data (Hochreiter & Schmidhuber, 1997; Bengio et al., 1994) .
In DS terms, this is generally due to the fact that flexible information maintenance over long periods requires precise fine-tuning of model parameters toward 'line attractor' configurations ( Fig. 1) , a concept first propagated in computational neuroscience for addressing animal performance in parametric working memory tasks (Seung, 1996; Seung et al., 2000; Durstewitz, 2003) .
Line attractors introduce directions of zero-flow into the model's state space that enable long-term maintenance of arbitrary values (Fig. 1) .
Specially designed RNN architectures equipped with gating mechanisms and (linear) memory cells have been suggested for solving this issue (Hochreiter & Schmidhuber, 1997; Cho et al., 2014) .
However, from a DS perspective, simpler models that can more easily be analyzed and interpreted in DS terms, and for which more efficient inference algorithms exist that emphasize approximation of the true underlying DS would be preferable.
Recent solutions to the vanishing vs. exploding gradient problem attempt to retain the simplicity of vanilla RNN by initializing or constraining the recurrent weight matrix to be the identity (Le et al., 2015) , orthogonal (Henaff et al., 2016; Helfrich et al., 2018) or unitary (Arjovsky et al., 2016) .
In this way, in a system including piecewise linear (PL) components like rectified-linear units (ReLU), line attractor dimensions are established from the start by construction or ensured throughout training by a specifically parameterized matrix decomposition.
However, for many DS problems, line attractors instantiated by mere initialization procedures may be unstable and quickly dissolve during training.
On the other hand, orthogonal or unitary constraints are too restrictive for reconstructing DS, and more generally from a computational perspective as well (Kerg et al., 2019) : For instance, neither
2) with flow field (grey) and nullclines (set of points at which the flow of one of the variables vanishes, in blue and red).
Insets: Time graphs of z 1 for T = 30 000.
A) Perfect line attractor.
The flow converges to the line attractor from all directions and is exactly zero on the line, thus retaining states indefinitely in the absence of perturbations, as illustrated for 3 example trajectories (green) started from different initial conditions.
B) Slightly detuned line attractor (cf.
Durstewitz (2003) ).
The system's state still converges toward the 'line attractor ghost ' (Strogatz, 2015) , but then very slowly crawls up within the 'attractor tunnel' (green trajectory) until it hits the stable fixed point at the intersection of nullclines.
Within the tunnel, flow velocity is smoothly regulated by the gap between nullclines, thus enabling arbitrary time constants.
Note that along other, not illustrated dimensions of the system's state space the flow may still evolve freely in all directions.
C) Simple 2-unit solution to the addition problem exploiting the line attractor properties of ReLUs in the positive quadrant.
The output unit serves as a perfect integrator, while the input unit will only convey those input values to the output unit that are accompanied by a '1' in the second input stream (see 7.1.1 for complete parameters).
chaotic behavior (that requires diverging directions) nor settings with multiple isolated fixed point or limit cycle attractors are possible.
Here we therefore suggest a different solution to the problem, by pushing (but not strictly enforcing) ReLU-based, piecewise-linear RNN (PLRNN) toward line attractor configurations along some (but not all) directions in state space.
We achieve this by adding special regularization terms for a subset of RNN units to the loss function that promote such a configuration.
We demonstrate that our approach outperforms, or is en par with, LSTM and other, initialization-based, methods on a number of 'classical' machine learning benchmarks (Hochreiter & Schmidhuber, 1997) .
More importantly, we demonstrate that while with previous methods it was difficult to capture slow behavior in a DS that exhibits widely different time scales, our new regularization-supported inference efficiently captures all relevant time scales.
In this work we have introduced a simple solution to the long short-term memory problem in RNN that on the one hand retains the simplicity and tractability of vanilla RNN, yet on the other hand does not curtail the universal computational capabilities of RNN (Koiran et al., 1994; Siegelmann & Sontag, 1995) and their ability to approximate arbitrary DS (Funahashi & Nakamura, 1993; Kimura & Nakano, 1998; Trischler & D'Eleuterio, 2016) .
We achieved this by adding regularization terms to the loss function that encourage the system to form a 'memory subspace', that is, line attractor dimensions (Seung, 1996; Durstewitz, 2003) which would store arbitrary values for, if unperturbed, arbitrarily long periods.
At the same time we did not rigorously enforce this constraint which has important implications for capturing slow time scales in the data: It allows the RNN to slightly depart from a perfect line attractor, which has been shown to constitute a general dynamical mechanism for regulating the speed of flow and thus the learning of arbitrary time constants that are not naturally included qua RNN design (Durstewitz, 2003; 2004) .
This is because as we come infinitesimally close to a line attractor and thus a bifurcation in the system's parameter space, the flow along this direction becomes arbitrarily slow until it vanishes completely in the line attractor configuration (Fig. 1) .
Moreover, part of the RNN's latent space was not regularized at all, leaving the system enough degrees of freedom for realizing arbitrary computations or dynamics.
We showed that the rPLRNN is en par with or outperforms initialization-based approaches and LSTMs on a number of classical benchmarks, and, more importantly, that the regularization strongly facilitates the identification of challenging DS with widely different time scales in PLRNN-based algorithms for DS reconstruction.
Future work will explore a wider range of DS models and empirical data with diverse temporal and dynamical phenomena.
Another future direction may be to replace the EM algorithm by black-box variational inference, using the re-parameterization trick for gradient descent (Kingma & Welling, 2013; Rezende et al., 2014; Chung et al., 2015) .
While this would come with better scaling in M , the number of latent states (the scaling in T is linear for EM as well, see Paninski et al. (2010) ), the EM used here efficiently exploits the model's piecewise linear structure in finding the posterior over latent states and computing the parameters (see Suppl. 7.1.3).
It may thus be more accurate and suitable for smaller-scale problems where high precision is required, as often encountered in neuroscience or physics.
7 SUPPLEMENTARY MATERIAL 7.1 SUPPLEMENTARY TEXT 7.1.1 Simple exact PLRNN solution for addition problem
The exact PLRNN parameter settings (cf. eq. 1) for solving the addition problem with 2 units (cf. Fig. 1C ) are as follows:

assistant: text: We develop a new optimization approach for vanilla ReLU-based RNN that enables long short-term memory and identification of arbitrary nonlinear dynamical systems with widely differing time scales.