Sparsity in Continuous-Depth Neural Networks
Authors: Hananeh Aliee
Till Richter
Mikhail Solonin
Ignacio Ibarra
Fabian J Theis
Niki Kilbertus
Venue: NeurIPS 2022 Accept (2022)
Keywords: Neural ODEs
interpretability
sparsity
dynamical systems

Abstract: Neural Ordinary Differential Equations (NODEs) have proven successful in learning dynamical systems in terms of accurately recovering the observed trajectories. While different types of sparsity have been proposed to improve robustness, the generalization properties of NODEs for dynamical systems beyond the observed data are underexplored. We systematically study the influence of weight and feature sparsity on forecasting as well as on identifying the underlying dynamical laws. Besides assessing existing methods, we propose a regularization technique to sparsify ``input-output connections'' and extract relevant features during training. Moreover, we curate real-world datasets including human motion capture and human hematopoiesis single-cell RNA-seq data to realistically analyze different levels of out-of-distribution (OOD) generalization in forecasting and dynamics identification respectively. Our extensive empirical evaluation on these challenging benchmarks suggests that weight sparsity improves generalization in the presence of noise or irregular sampling. However, it does not prevent learning spurious feature dependencies in the inferred dynamics, rendering them impractical for predictions under interventions, or for inferring the true underlying dynamics. Instead, feature sparsity can indeed help with recovering sparse ground-truth dynamics compared to unregularized NODEs.

Sparsity in Continuous-Depth Neural Networks
Hananeh Aliee
Till Richter Helmholtz Munich
Mikhail Solonin" Technical University of Munich
Helmholtz Munich
Ignacio Ibarra Helmholtz Munich
Fabian Theis
Niki Kilbertus
Technical University of Munich Helmholtz Munich
Technical University of Munich Helmholtz AI, Munich
(hananeh. aliee, till. aille,till.eee...eee.....iil will.richterr........ richter, ignacio. ibarra, fabian. theis, theis,niki kilbertus) @helmhol wwathoolzzmmueccenncco
Abstract
Neural Ordinary Differential Equations (NODEs) have proven successful in learn- ing dynamical systems in terms of accurately recovering the observed trajectories. While different types of sparsity have been proposed to improve robustness, the generalization properties of NODEs for dynamical systems beyond the observed data are underexplored. We systematically study the influence of weight and fea- ture sparsity on forecasting as well as on identifying the underlying dynamical laws. Besides assessing existing methods, we propose regularization technique to sparsify "input-output connections" and extract relevant features during training. Moreover, we curate real-world datasets consisting of human motion capture and human hematopoiesis single-cell RNA-seq data realistically analyze different levels of out-of out-of-distribution (OOD) generalization in forecasting and dynamics identification respectively. Our extensive empirical evaluation on these challenging benchmarks suggests that weight sparsity improves generalization in the presence noise or irregular sampling. However, it does not prevent learning spurious feature dependencies in the inferred dynamics, rendering them impractical for pre- dictions under interventions, or for inferring the true underlying dynamics. Instead, feature sparsity indeed help with recovering sparse ground- truth dynamics compared to unregularized NODEs.
Introduction
Extreme wver-parameterization has shown to be part of the success story of deep neural networks [52, [5] 4]. This has been linked to evidence that warrpaarametriize models are easier to train,
perhaps because they develop "convexity-like" properties that help convergence of gradient descent
10 [4] However, over-parameeterization comes at the cost of additional computational footprint during training and inference [3] Therefore, motivation for sparse neural networks manifold and include: (i) imitation of human learning, where neuron activity is typically sparse (2). (ii)
computational efficiency (speed up. memory reduction, scalability) 200] (iii) interpretability [51 as
well as (iv) avoiding overfitting or improving robustness [6].
Enforcing sparsity in model weights, also called model pruning, and its effects for standard predictive modelling tasks have been extensively studied in the literature [20]. However, sparsity in continuous- depth neural nets for modeling dynamical systems and its effects generalization properties is still
Work done while at TUM. MS is currently employed by J.P. Morgan Chase & Co.; mikhail. solonin @ jpmorgan.com
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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underexplored. Neural Ordinary Differential Equations (NODEs) [T] have been introduced as the limit of taking the number of layers in residual neural networks to infinity, resulting in continuous- depth (or continuous-time) neural nets. NODEs were originally predominantly used for predictive tasks such as classification or regression, where they learn some dynamics serving the predictive task. These networks have also shown great promise for modeling noisy and irregular time-series. More recently, it has been stated that NODEs can also be thought of as learning the underlying dynamics (i.e., the ODE) that actually govern(s) the evolution of the observed trajectory. In the prior scenario (predictive performance), large body of literature has focused on regularizing the number of model evaluations required for single step of the ODE solver for improving the efficiency 16 22 39 18 38). These regularization techniques rely on learning one out many possible equivalent dynamical systems giving rise to the same ultimate performance, but are easier and faster to solve. Crucially, these regularizers do not explicitly target sparsity in the model weights or in the features used by the model.
In the latter scenario (inferring dynamical laws), we as- sume the existence of a ground truth ODE that governs the dynamics the observed trajectories. Here, two types of sparsity may be of interest. First, motivated by stan- dard network pruning, sparsity in model weights (weight sparsity or model sparsity) can reduce computational re- quirements for inference [28]. see Figure top). Second, it has been argued that feature sparsity, i.c., reducing the number of inputs a given output of the NODE relies on, improve identifiability of the underlying dynamical law B 7 30), see Figure [bottom). The motivation for feature sparsity often relies on an argument from causality, where it is common to assume sparse and modular causal dependencies between variables 45 52
weight spansty
Artitary architecture
Outputs
XP
XB
12
X2
X3
X3
festure sparsity
Inputs
Arbitrary architacturs
Oulput
In this paper, we empirically assess the impact of various types of eparsity-efforeingg methods on the performance of NODEs on different types of OOD generalization for both prediction and for inferring dynamical laws on real- world datasets. While most existing methods target weight sparsity, highlight that feature sparsity can improve interpretability and dynamical system inference by propos- ing new regularization technique for continuous-time models. Specifically, our main contributions are the following:
Figure I: Model vs feature sparsity
We propose RathReg differentiable LO-based regularizer that enforces sparsity of "input-output paths" and leads to both feature and weight sparsity.
We extend L0 regularization [31] and LassoNet (26) to NODEs and perform extensive comparisons
with existing models including vanilla NODE [T]. C-NODE (3) GroupLasso for NODEs and our PathReg.
We define useful metric for evaluating feature sparsity, demonstrate the differences between weight and feature sparsity, and explore whether and when one implies the other.
To cover at least part of the immense diversity of time-series problems, each of which implying different types of OOD challenges, curate large, real-world datasets consisting of human motion capture (mocap. cs. cmu. edu) as well as human hematopoiesis single-cell RNA-seq [32] data for our empirical evaluations.
2
Background
2.1
Continuous-depth neural nets
Among the plethora of deep learning based methods to estimate dynamical systems from data
29 44
27 38), we focus on Neural Ordinary Differential Equations (NODEs), In NODEs, a neural
network with parameters 0 is used learn the function fe x f from data, where f defines (first
'he python implementation available at: https:/giitab.ccmmm/iiiib//apttt

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order) ODE in its explicit representation X - f(X,t) [T Starting from the initial observation X (a) some time = a, an explicit iterative ODE solver is deployed to predict (t) for E (a. bj using the current derivative estimates from So- The parameters 0 are then updated via backpropagation on the mean squared error (MSE) between predictions and observations. As discussed extensively in the literature, NODEs can outperform traditional ODE parameter inference techniques in terms of reconstruction error, especially for non-linear dynamics f [S]. In particular, one advantage of NODEs over previous methods for inferring non-linear dynamics such as SINDy [9] dictionary of non-linear basis functions has to be pre-specified.
is that no
A variant of NODEs for second order systems called SONODE exploits the common reduction of higher-order ODEs to first order systems [37]. The second required initial condition, the initial velocity X (a), is simply learned from (a) via another neural network in an end-to-end fashion. Our experiments build on the public NODE and SONODE implementations. However, our framework readily applicable to most other continuous-depth neural nets including Augmented NODE [[5]
latent NODEs
44
and neural stochastic DEs [27 48
2.2 Sparsity and generalization
Two prominent motivations for enforcing sparsity in over-parameterized deep learning models are (a) pruning large networks for efficiency (speed, memory). (b) as a regularizer that can improve interpretability or prevent overfitting. In both cases, one aims at preserving (as much as possible) i..dd generalization performance, i.e., performance on unseen data from the same distribution as the training data, compared to a non-sparse model [20].
Another line of research has explored the link between generalizability of neural nets and causal learning [45). where generalization outside the i.i.d. setting, is conjectured to require an underlying causal model. Deducing true laws of nature purely from observational data could be considered an instance of inferring a causal model of the world. Learning the correct causal model enables accurate predictions not only the observed data (next observation), but also under distribution shifts for example under interventions [45]. A common assumption in causal modeling is that each variable depends on (or is a function of) few other variables [45 54
g In the ODE context, we interpret the variables (or their derivatives) that enter a specific component /s as causal parents, such that we
36
can write X, = ((pa(X.). t) [35 Thus, feature sparsity translates to each variable X; having only few parents pa(Xx, (x.... which can also be interpreted as asking for "simple" dynamies. Since weight sparsity as well as regularizing the number of function evaluations in NODEs can also be considered to bias them towards "simpler dynamies", these notions are not strictly disjoint, raising the question whether one implies the other.
In terms of feature sparsity, Aliee al. 3. Bellot et al. [7] study system identification and causal structure inference from time-series data using NODEs. They suggest that enforcing sparsity in the number of causal interactions improves parameter estimation as well as predictions under in- terventions. Let us write out So as a fully connected net with L. hidden layers parameterized by 0 = (W1, b L-1 as
fo(X) = WW:+ (l))= o oWW X + +b')+b))...
with element-wise activation function a,, [-th layer weights W and biases b'. Aliec et al. [3 then seek to reduce the overall number of parents of all variables by attempting to cancel all contributions of a given input on a given output node through the neural net. In a linear setting, where (ir) = x, the regularization term is defined by
|A = || II..
(2)
where A, = O if and only if the i-th output is constant in the j-th input. In the non-lineat setting, for certain (xx) &f x, the regularizer ||A|]].1 ||| III (with entry wise absolute values on all W is an upper bound on the number of input-output dependencies, i.e., for each output i summing up all the inputs j it is not constant in. Regularizing input gradients 42 4 is another alternative to train neural networks that depend on fewer inputs, however it is not scalable to high-dimensional regression tasks
'The 1, norm || I|.1 is the sum of absolute values of all entries of W.
3

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Bellot et al. 7 instead train separate neural net fo. RR" - R for each variable X; and penalize
NODEs using GroupLasso on the inputs via
(3)
R1
where W. is the weight matrix in the input layer of fi and [W]]., refers to the hth column of W! that should simultaneously (as group) be set to zero not. While enforcing strict feature sparsity (instead of regularizing an upper bound), parallel training of multiple NODE networks can be computationally expensive (Figure bottom). While this work suggests that sparsity of causal interactions helps system identification, its empirical evaluation predominantly focuses on synthetic data settings, leaving performance on real data underexplored.
Another recent work suggests that standard weight or neuron pruning improves generalization for NODEs (28). The authors show that pruning lowers empirical risk in density estimation tasks and decreases Hessian's eigenvalues, thus obtaining better generalization (flat minima). However, the effect of sparsity identifying the underlying dynamics as well as the generalization properties of NODEs to forecast future values are not assessed.
3 Sparsification of neural ODEs
In an attempt to combine the strengths of both weight and feature sparsity, we propose a new regularization technique, called PathReg, which compares favorably to both C-NODE [3] and GroupLasso 77. Before introducing PathReg, we describe how to extend existing methods to NODEs for exhaustive empirical evaluation.
3.1 Methods
LO regularization. Inspired by 31, we use a differentiable LO norm regularization method that can be incorporated in the objective function and optimized via stochastic gradient descent. The LO regularizer prunes the network during training by encouraging weights to get exactly zero using set of non-neeee -eggatiee stochastic gates z. For an efficient gradient-based optimization, Louizos et al. [31) propose to use a continuous random variable with distribution g(s) and parameters é, where is then given by
q(s Φ),
min(1, max(0, s)).
Gate = is hard-sigmoid rectification of 2² that allows the gate to be exactly zero. While we have the freedom to choose any smoothing distribution gs)), we use binary concrete distribution [33 21 as suggested by the original work (3T]. The regularization term is then defined the probability of being positive
x - ((s - 6)
(5)
where Q is the cumulative distribution function s. Minimizing the regularizer pushes many gates to be zero, which implies weight sparsity these gates multiplied with model weights. LO regularization can be added directly to the NODE network fa-
LassoNet is feature selection method [26] that uses input-to-outphy skip (residual) connection that allows a feature to participate hidden unit only its skip connection is active. LassoNet can be thought of as residual forward neural net Y S x + he (X), where he denotes another ferd-forward network with parameters 6, S E g=xn refers to the weights in the residual layer, and Y the responses. To enforce feature sparsity, LI regularization is applied to the weights of the skip connection, defined as ||S|],,1 (where I| lh, denotes the element- wise LI norm). A constraint term with factor P is then added to the objective to budget the non-linearity involved for feature ie relative to the importance of X₃
min (0, + S||S||1 8.5
subject to
|| S Pl[[S]. for k E ......,),
(6)
where [ denotes the hth column of the first layer weights of hg, and SS.. represents the 4th column of S. When / = O, only the skip connection remains (standard linear Lasso), while p - oo corresponds to unregularized network training.

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weights
paths
10a
Init values
values after
log (c(cossss)
logicoums)
107
10
10a
-0.4
-0.2
0.0
0.2
0.4
0.0
0.5
1.0
1,5
2.0
2.5
3.0
Figure 2: The distributions of network weights and paths weights (over the entries of the matrix in Eq. [3] using PathReg applied to single-cell data in Section 4.3 feature sparsity.
PathReg increases both model and
LassoNet regularization can be extended to NODEs by adding the skip connection either before or after the integration (the ODE solver). If added before the integration, which we call Inner LassoNet, linear function of X is added to its time derivative
X S x + fo(X,t), X(0) = 20
(7)
Adding the skip connection after the integration (and the predictor o), called Outer LassoNet, yields
X, S - + oOODESolver(fa. Zo, to.
PathReg (ours). While LO regularization minimizes the number non-zero weights in a network, does not necessarily lead to feature sparsity meaning that the response variables can still depend on all features including spurious ones. To constrain the number of input-outpul paths, i.e., to enforce feature-sparsity,
regularize the probability of any path throughout the entire network contributing to an output. To this end, we use non-negative stochastic gates = g(s) similar to Eq. where probability of an input-output path P being non-zero is given by
q(P + 0) = II ((= 0 σ)
zEP
and we constrain
#paths Σ E
II SEP;
(10)
to minimize the number of paths that yield non-zero contributions from inputs to outputs. This is equivalent regularizing the gate adjacency matrix A, Gl++1 GI++).....G G¹. Where G' is probability matrix corresponding to the probability of the I-th layer gates being positive. Then, Az,, represents the sum of the probabilities of paths between the i-th input and the j-th output. Ultimately, we thus obtain our PathReg regularization term
1All1,,1 || ...... with = + 0 (1)).
(11)
where (2(2 * 0) with parameters Φi,j is the probability of the 1-th layer gate Zij being nonzero. Regularizing I| minimizes the number of paths between inputs and outputs and induces no shrinkage on the actual values of the weights. Therefore, we can utilize other regularizers 0 such as ||||),, 1 in conjunction with PathReg similar to Eq, In this work, we consider the following overall loss function
N Σ
c(olf(ER00 ((s))) o(s)). vi)) + AallAA + A |Allaa
R(0, Φ) = Eg(()())
N
(f. Φ) + Ao 1111hhhh + A111-111,1
(12)
where L corresponds to the loss function of the original predictive task, L the overall predictive loss. is the NODE-hased model with / modeling time derivatives, g(-) == min(1, max(0, ))), Ao A 2 regularization parameters, and @ is entry-wise multiplication. Lg measures how well the model
5

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Table 1: Results for the synthetic second-order ODE.
MSE
SPARSITY (%)
MODEL.
TRAIN
EXTRAPOLATION
FEATURES
WEIGHTS
BASE
1,3 X 10
3.2 X 10-3
47.9
11.1
LO
1.9 X 10
3.3 x 10
0.0
49.6
C-NODE
6.6 10
4.1 x 10
75.5
16.7
PATHREG
6.3 x 10
3.9 x 10
75.5
35.2
LASSONET
8.4 10
x 10
0.0
0.0
GROUPLASSO
8.8 x 10
1.5 x 10
61.1
6.0
fits the current dataset. While || shrinks the actual values of weights 0 in an attempt to zero out entire paths, ||A:||1.1 enforces exact zeros for entire paths at once. When As., the output is constant in input j. In practice, during training the expectation in Eq. 12 estimated usual via Monte Carlo sampling.
Unlike GroupLasso 77 PathReg does not require training of multiple networks in parallel. Moreover, unlike C-NODE [3) PathReg leads exact zeros in the weight matrices and requires no choice of threshold for deciding which paths are considered as zeros (see Figure 2
3.2 Training sparse models
Sparsity can generally be enforced during after model training. In this paper, we study main approaches [20] to train sparse models: (i) sparse training, where we add regularizers already during model training; (II) training and iterative sparsification, where iteratively increase regularization parameters only after the model has been trained (close to) to convergence. A detailed description of our training procedures and architecture choices for each experiment is provided in Appendix
3.3 Assessing feature sparsity
For models regularized with LO or PathReg the natural measure for feature sparsity is
(W>++) Gh++) .... (WW' & G¹);
(13)
where absolute values are taken entry-wise LassoNet also allows for natural feature sparsity metric by counting the non-zero input weights Σ ||W;"]|2 > e of the first hidden layer, where denotes the number of input features and is small tolerance parameter. To evaluate feature sparsity for other feed-forward continuous-depte models, we count the number of non-zero elements W and within some tolerance
A | ... [ again with entry wise absolute values (set to le 5 in our experiments). This amounts to an upper bound on the number of input-output dependencies: while we can have A., * 0 even though X, pa (X.), A 0 always implies X, pa (X,).
4 Results
In this section, we present empirical results on several datasets from different domains including human motion capture data and large-scale single-cell RNA-seq data. We demonstrate the effect of the different sparsity methods including LO [1], C-NODE B., LassoNet [26). GroupLasso [7]. and PathReg the generalization capability of continuous- aspects of generalization by assessing the accuracy of these models for time-serics forecasting and the identification of the governing dynamical laws. In all examples, assume that the observed motion can be (approximately) described by a system of ODEs where variables correspond directly to the observables of the trajectories.
neural nets. We address different
4.1 Sparsity for system identification
motivate sparsity for NODEs starting with a simple second-order NODE as used by Aliee et al. 3 The differential equations explaining this system presented in Appendix B. The questions

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Method
Overall
Sparsities
Panking
Best
Base
L0
Model
Wios
C-NODE
Model
Perfommance (MS1)
PathReg
Model
Feature
Warsi 8.50-3
2.0p-4
LassoNat Festure
Reg Reguterzae
ap
Ovenl
Soore
Oreral
Trai
Walk
Train
Golf
Than
Wavi
Test
Walk
Toat
Oreral
weight
Fashure
Min
Max
D.D
84.33
Figure 3: Qualitative result comparison for the human motion capture data. Circle sizes indicate the scaled (continuous) metrics: MSE (blue) and sparsity (in %) (green). The lengths of bar show the mean metric over all sparsities / performances for a method. Different shades the same color indicate the ranking from best to worst method within given column. The overall score (orange) the average over all the scores, including both performance and sparsity, for each method across datasets. We then rank the methods accordingly. PathReg yields the best overall results in this comparison.
that we try to answer here are whether different sparsity methods can (i) learn the true underlying system, and (ii) forecast into the future (extrapolate). For this linear system, the adjacency matrix represents the exact coefficients in Eq. [4 The results summarized in Tables [ PathReg, C-NODE, LassoNet, and GroupLasso perform well with respect to reconstruction error. Moreover, PathReg, C-NODE, and GroupLasso lead to higher sparsity and outperform other methods for extrapolation. Among these methods, only PathReg and C-NODE able learn the true system with the lowest erro
and 2 show that
[4
4.2 Sparsity improves time-series forecasting
We next demonstrate the robustness of sparse models for both reconstructing and extrapo- lating human movements using real motion capture data from mocap. cmu. Each datapoint is 93-dimensional (31 joint locations with three dimensions each) captured over time. We select three different movements including walking, waving, and golfing, and use 100 frames each for training. After training, we query all models to extrapolate the next 100 frames. For walking and golfing, where multiple trials are available, we also test the model on unseen data in the sense that it has not any subset of those specific se- quences (despite having trained on other sequences depicting the same type of movement). Generally, for time-series forecasting, it is not straight forward to give precise definitions of in-, and fittof--disribution. In Table 7 in Ap- pendix C we show a grid of plots illustrating loss and sparsity ization parameter of each method on this dataset. We show detailed results in Table 3 in Appendix [[]and summarize them concisely in Figure 3] We did not manage scale Grou- pLsss to this 93-dimensional dataset. While all other models show comparative performance (in terms of MSE error), only PathReg achieves strong levels of both weight and feature sparsity (quantitative comparison is in Table 3.. The re- sults show that PathReg outperforms other meth-
odtipoation
trem comn
function of the regular-
Bmm
& A.A.A
A
&
&
$ 8
$3
PutiReg
Figure 4: Human-movement reconstructions and extrapolation.
Caooupassso is originally designed for first-order ODEs. Here, we use an augmented version of that feeding also the initial velocities to the network. Therefore, direct comparison with its adjacency matrix is not entirely adequate.
7

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type
dot pacadotime
G31é= Mend
Monocyte lineage
C316+
Enytrobiest
GIM pig
YEC
ymmp prog
MIEE prog
lineage
Nalue CD20+
Procrytirallic
Transitonal
Enythroid linage
C-NODDE
2330
5121-
5330
500
2220
$17-
RathReg
0330
10%
02
Figure 5: Single-cell RNA-seq Data. Shown are 2D UMAP visualisation of training data colored by cell types and pseudotime (top) where each point is a cell, and the expressions of three genes and their predictions over pseudo time using C-NODE (top) and PathReg (bottom). The predictions across all genes are presented Figure
ods with respect to MSE for both train and test datasets while resulting in the highest model and feature sparsity.
We further examine the extrapolation (in time) capability of these models. We observe that all sparsity regularization techniques improve extrapolation over unregularized NODES. Figure qualitatively illustrates the reconstruction and extrapolation performance of both the baseline and PathReg for the walking dataset (visually, the results are similar to PathReg for other regularizers). While these models perform well for extrapolation and generalization to unseen data, we show that only PathReg and C-NODE are able to avoid using spurious features for forecasting. Figures and in Appendix C show that PathReg and C-NODE learn sensible and arguably correct adjacency matrices (derived from joint connectedness). whereas L.0 and LassoNet fail to sparsify the input-output interactions appropriately (more details in Appendix C.2 Since we use second-order NODEs for this dataset, the
adjacency matrix described in Section 3.3 represents how the acceleration predicted by fo of each
joint point (e.g., X,) depends on the positions as well as the velocities of other joint points (e.g., Xz). In case the dependency is non-constant, we infer that the joint point X; interacts with Xj.
4.3 Sparsity improves dynamic identification for single-cell data
Finally, we explore how pruning (i) improves real-world generalization and (ii) helps learn gene- gene interactions in human hematopoiesis single-cell multiomics data (32) (GEO accession code GSE194122). The regulatory (or causal) dependencies between genes can explain cellular functions and help understanding biological mechanisms []. Gene interactions are often represented as gene regulatory network (GRN) where nodes correspond genes and directed edges indicate regula- tory interactions between genes. Reliable GRN inference from observational data is exceptionally challenging and still an open problem in genomics
40) As human regulatory genes known as
transcription factors (~ 1,600 (25),, do not individually target all genes (potentially up to ~ 20,000), the inferred GRN is expected to be sparse. Thus, predicting the expression of gene should depend on few other genes that can be restricted using sparsity-enforeing regularizers.
8

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Method
Overall
Performance
Sparsities
Far king
Best
Base
Model
C-NODE
Model
Performance (MSE)
PathReg
Model
Feature
Worst 1.58
0.005
Best
LassoNet Feature
Sparsity (H4
Name
Reguar
appled
Ovaall
Score
Oniaal Parormmns
S1D1
SENN
SS06
Overall Spprreeee
Wiigit
Festure
80
Figure 6: Comparison results using the single-cell RNA-seq Data. PathReg shows best overall results, as well as with regard to performance, and to sparsity. The models are trained with batch S1D2, and tested using other four batches.
For assessing real-world generalization of various sparsity-enforcing methods, we use human bone marrow data of four diverse donors, generated at three different sites using five sequencing batches (the pre-processing steps applied to the data are discussed in Appendix [B3).. In this experiment, we train NODE models with one batch (SID2), and later test those using other four batches (SID1, S2D1, S2D4, and S3D6). Notably, time-resolved ground truth dynamics of cells throughout their lifetime are not available, because all existing genome-wide technologies are destructive to the cells [32. Therefore, scRNA-seq only offers static snapshots of data and the temporal order of cell states is commonly estimated based on certain distances between their measured transcriptomes. Following that, we first apply the diffusion pscudo-time (DPT) ordering approach [19) diffusion map based manifold learning technique, to rank the cells. This allows to generate a cell fate mapping from hematopoietic stem cells (HSC) into three biological lineages that used for training: Monocyte, Erythroid, and B cells (Figure5) NODEs are then trained using highly- variable genes (529 genes) with time taken to be the inferred pseudo-time (Figure 9
The train and test MSE across 5 batches and three pruning methods are presented in Table and 7 in Appendix C..3 and summarized in Figure While all regularized models achieve decent predictive performance, PathReg outperforms the other methods in most scenarios. We further study the cunnsription--fctor-t----- transcription factors (columns) and target genes (rows) (see Appendix [B.3] for more details). The adjacency matrix A demonstrates what transcription factors (on columns) are used for predicting the expression value of a target gene (on rows). [Aiyl > for a small threshold c = le 5 can then be interpreted interaction between gene and gene j being present in the data (see Figure I0 for an illustrating example). For visualisation purposes, Figure only shows the adjacency matrices across the Erythroid lineage. However, the results are comparable for other lineages. The used list of transcription factors and target genes are extracted using chromatin and sequence-specific motif information (we again refer to Appendix [B.3] for more details). Since ground truth dynamics known for GRNs, validating the inferred interactions for multiple cell types is a challenging task. However, we observe that PathReg and LO are able remove the interactions between pairs known to not directly causally influence each other, including target genes (called spurious features in Table 5 and further discussed in Appendix B.3). while the other models fail in that regard. Assessed by the literature, we also observe that known Erythroid transcription factors are inferred as important by both PathReg and LO (RUNXI 24 ETV6 44), NFIA 34 MAZ 12), GATAI (53).
gene interactions via the adjacency matrices filtered for
Conclusion and discussion
We have shown the efficacy of various sparsity enforcing techniques on generalization of continuous- depth NODES and proposed PathReg, a regularizer acting directly on entire paths throughout a neural network and achieving exact zeros. We curate real-world datasets from different domains consisting of human motion capture and single-cell RNA-seq data. Our findings demonstrate that sparsity improves
generalization (for the types of OOD considered) of NODEs

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when our objective is system identification from observational data, extrapolation of a trajectory. or real-world generalization unseen biological datasets collected from diverse donors. Finally, we show that unlike weight sparsity, feature sparsity as enforced by PathReg can indeed help identifying the underlying dynamical laws instead of merely achieving strong in-distribution predictive performance. This is particularly relevant for applications like gene-regulatory network inference, where the ultimate goal not prediction and forecasting, but revealing the true underlying regulatory interactions between genes of interest. We hope that our empirical findings as well as curated datasets can serve as useful benchmarks to a broader community and expect that extensions of framework to incorporate both observational and experimental data will further improve practical system identification for ODEs. Finally, our path-basce regularization technique may be of interest to other communities that aim at enforcing various types of shape constraints or allowed dependencies into deep learning based models 5
Acknowledgments and Disclosure of Funding
We thank Ronan Le Gleut from the Core Facility Statistical Consulting at Helmholtz Munich for the statistical support. HA and FJT acknowledge support by the BMBF (grant O1IS 18036B) and the Helmholtz Association's 's Initiative and Networking Fund through Helmholtz AI (grant ZT-I-PF-5-01) and CausalCellDynamiic (grant Interlabs-0029), TR is supported by the Helmholtz Association under the joint research school "Munich School for Data Science" MUDS. NK was funded by Helmholtz Association's 's Initiative and Networking Fund through Helmholtz AL FJT reports receiving consulting fees from Roche Diagnostics GmbH and ImmunAI, and ownership interest in Cellarity, Inc. and Dermagnostix. The remaining authors declare no competing interests.
References
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14


Reviews:
[{"review_id": "xaLxDhr2W5", "reviewer": "Reviewer_ii69", "summary": "This work proposes modifications (suggestions) to improve the performance of the Neural Ordinary Differential Equations towards better generalization. Specifically, for the neural network based function in X’=f(X, t), they propose a regularization technique for the weights of `f’ based on a combination of path-norm and L0 norm. The proposed regularization targets inducing feature sparsity, which ensures that the output variable depends on a few number of parents X_i’ = f(pa(X_i, t) and also the weight sparsity, which is traditionally shown to help with the generalization capability of the neural networks. This work primarily introduces a differentiable L0-based regularizer (PathReg) and extends it to NODEs. The authors argue empirically that the technique proposed by them improves over the state-of-the-art methods over a range of time-series tasks. ", "questions": "Questions:\n1. (Line 175) $W_j^(1)$ represents the first layer weights of $h_\\theta$ ?\n2. What are the choices of the distribution `q’ used for the L0 reg? \n3. I understand that the equation 10 & 11 enforce the same constraint. In eq. (11), can you please expand on the form of the distribution $q^l( z | \\phi)$ ? I believe that the $\\phi$ params will be learned during the training process. \n4. In equation (12), the expectation term samples `s’, as per my understanding, gives a set of G’s. Is the ||A_z|| term changing with every sample? How does the $\\phi$ term come into picture here? \n\nI request the authors to make these above points clearer in the paper as well. I went through the supplementary material too but I was not able to find clear descriptions of the procedure. \n\nExperiment suggestions:\n5. The \\lambda_1 ||A||_1 term is introducing weight sparsity. How does the performance change if this term is dropped and only \\lambda_0 ||A_z||_1 is used? \n", "limitations": "N/A", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "(previous work summary) C-NODE uses the upper bound of L1(path-norm) and GroupLasso uses the group lasso norm as regularization. LassoNet introduces a skip connection and then enforces sparsity by putting constraints on the skip connection weights and the first layer of the feedforward NN. \n\nBefore I can assess the quality, novelty and impact of the work, I request the authors to address the points raised in the Questions & suggestions section. I am not able to follow certain aspects of the work. \n\nUpdate: After the clarifications provided by the authors, I have updated my ratings.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "klHaBn0psAt", "reviewer": "Reviewer_388f", "summary": "This paper focuses on identifying and empirically verifying the ways that Neural ODEs can be regularized towards improved generalization. Sparsity in weights and features are explored in particular, where a new regularization technique called PathReg is developed as an analogue to other common regularization techniques in deep neural network training. There are experiments performed on mocap and RNA-sequencing datasets to assess the generalization capabilities of continuous-depth neural networks.\n", "questions": "Do these changes lead to more favorable ODE solvers over others? What are the speedups in general?\n\nWould you be able to compare again other neural ODE techniques and summarize their differences.\n", "limitations": "Not discussed.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Novelty, relevance, significance\nThe work is a thorough empirical survey of the proposed PathReg method which is a unification of L0 regularization (which specifying the weights in a neural net) and LassoNet (which selects features based on how much an input affects the output) with residual weight regularization. The loss is modified in the usual way to include these extra regularization terms and generally the results point at the fact that combining feature and model selection, building upon improvements shown by either methods already achieved independently. \n\nI generally find data augmentation types of papers difficult to rate in terms of high novelty. It also makes sense that better out-of-distribution generalization is achieved when the model is not trained to over-fit to in distribution and hence can better ignore spurious features.\n\n## Soundness / Correctness & Clarity\nThe derivations for the new loss and adaptation of existing two methods for regularization / sparsification are generalized correctly to the NODEs setting. L0 regularization of terms is also standard and hence sound.\n\n## Quality of writing/presentation\nWriting and presentation are well-done. Figures are clear and labeled.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "MIBgVoNKmeQ", "reviewer": "Reviewer_hJba", "summary": "In this work, the authors propose a novel feature regularization method called PathReg for NODEs, a neural net architecture that can reconstruct and extrapolate time series data. This method allows the NODE to learn parameters with 2 types of sparsity: 1) weight sparsity and 2) feature sparsity. While methods exist to regularize NODEs with feature sparsity (C-NODE and Group Lasso), there is currently no approach that also integrates weight sparsity. The primary contribution of this work is to first adapt two methods for weight sparsity in regular neural networks to NODEs (L0 and LassoNet) and then combine both types of sparsity in PathReg. The authors evaluate their method on a set of convincing simulated and real datasets against several baselines using reasonable metrics. Furthermore, they provide some interesting insight on the data by interpreting the feature correlations derived from their model. The results and experiments described in the manuscript are quite promising. However, there are several claims that need to be clarified and additional experiments that would be needed to support claims or \"nice to have\" to improve the paper. Details are given below.", "questions": "See above.", "limitations": "Yes", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "(1) \nFirst, we very much appreciate the author’s thorough evaluation of the existing literature in this space and the clear statement of their contribution. It was easy to recognize exactly how the author’s are uniquely innovating in the context of other approaches in the field. \n(2)\nThe author’s describe several advantages of sparsity. These include: 1) improved speed and efficiency, 2) better generalization, and 3) improved interpretability. \n(a)\nWould it be possible to show runtime benchmarks for each method? The reason for requesting this benchmark is that C-Node performs comparably (and sometimes better on certain datasets) on the reconstruction metrics. One reason to justify the use of PathReg would be the runtime. Is it the case that learning fewer weight parameters by combining both feature and weight regularization makes the approach faster? \n(b)\nIs it possible to perform a similar interpretation on the A matrix in C-NODE as was done on the A matrix in PathReg? If not, can the interpretability of C-NODE and PathReg results be compared in some other way?\n(3)\nIt is not completely clear what the overarching premise of combining both feature and weight sparsity actually achieves in practice. In other words, why should one care that the weight metric is higher for PathReg in cases when the generalization error is comparable. \n(4)\nIt is not clear what Method Ranking and Overall Score mean in Figure 3 and 6. Regardless, these are not useful metrics and could be discarded to make the figures more clear. \n(5)\nIt is not clear what path weight means (x-axis in right most plot in Figure 2).\n(6)\nThe interpretation of the correlation matrix A for both the motion capture data and single cell data are quite exciting. We would encourage the authors to consider adding some examples to the main text and to elaborate on subjective claims such as “learn sensible and arguably correct adjacency matrices (derived from joint connectedness)”. For example, were leg and arm joints more connected in the walking data (where both appendages move) compared to the golf data (where legs do not move). In general, we did not find Figure 4 or Figure 5 (top) to be particularly useful for illustrating the main claims in the paper and could be replaced.\n(7)\nThe C. elegan embryo development dataset in Packer et al. 2019 has single cell time series data with experimentally derived time. This could be used instead of the inferred pseudotime. The approach used to infer pseudotime is also not currently state of the art. Please refer to Saelen’s et al. 2019 for thorough benchmarks of over 70 methods. \n(8)\nRNASeq batches should be corrected for batch effect. Consider methods like COMBAT or RUVSeq. \n(9)\nIt is not clear how spurious features are defined in Figure 6. How did the authors determine that a pair of genes did not directly affect each other’s expression? What does it mean for a transcription factor to be inferred by PathReg as “important”? One approach to determine whether certain TFs affect other genes is to take knockout data (such as from ENCODE) to perform differential expression between control and TF KO data.\n(10)\nPlease elaborate on the contribution for releasing data. What aspect of the data was modified to warrant re-release under this publication? \n(11)\nThe Base model has significantly worse generalization error compared to other methods for gene expression. This was not so much the case with the motion data where the generalization error was comparable to the other methods for some datasets. This is quite an interesting result. Why do the authors think this is the case and can their model’s improved interpretability aid in figuring out the answer?  \n\nMinor errors and comments:\n-----------------------------------------\nIn the Figure 1 feature sparsity figure, shouldn’t x0 in the input be connected to x0 in the output? How can it be possible for a feature to have no effect on reconstructing itself? Even though this is a cartoon, this is a bit misleading.\n\nTypo line 241: PsthReg -> PathReg\n\nPerhaps highlight that S1D2 was the training data in Figure 6\n\nIs the feature sparsity % of exactly 75.52 duplicated multiple times in Table 1 correct? \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "FxzZKnef0h7", "reviewer": "Reviewer_Mra2", "summary": "The authors propose a method PathReg to improve sparsity in NODE models. The authors backup their claim that PathReg improve sparsity and performance by experiments in 3 different datasets.", "questions": "- The design of PathReg seems to be fairly straight forward with feed-forward networks. What is the reason you choose NODE-based models over general feed-forward networks?\n- Gating are dependent on initialization so probably multiple experimental runs are needed for an accurate evaluation. How many times have you run each of your experiments?\n- Gating could also potentially slow down the training process. What is the time consumption compared to other models?", "limitations": "As far as I know, there will hardly be any negative societal impact of the work.", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- The figures are nice and well-represent the experiments\n- The effect of sparsification of PathReg is significant\n- The performance of PathReg is descent. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept