Learning invariant representations of time-homogeneous stochastic dynamical systems
Authors: Vladimir R Kostic
Pietro Novelli
Riccardo Grazzi
Karim Lounici
Massimiliano Pontil
Venue: ICLR 2024 poster (2024)
Keywords: Dynamical systems
statistical learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

LEARNING INVARIANT REPRESENTATIONS OF TIME- HOMOGENEOUS STOCHASTIC DYNAMICAL SYSTEMS
Vladimir R. Kostic"
Pietro Novelli"
Istituto Italiano di Tecnologia University of Novi Sad vladimir. www.kossiiiiiic
Istituto Italiano di Tecnologia pietro. wwweellliii....
Riccardo Grazzi
Karim Lounici CMAP École Polytechnique
Massimiliano Pontil
Istituto Italiano di Tecnologia University College London
Istituto Italiano di Tecnologia University College London
ABSTRACT
We consider the general class of time-honee hom--ooooogoeu stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function propose is associated with projection operators from the representation space the data space, overcomes metric distortion, and can be empirically estimated from data, In the discrete-time setting, we further derive relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.
INTRODUCTION
Dynamical systems are mathematical framework describing the evolution of state variables over time. These models, often represented by nonlinear differential equations (ordinary or partial) possibly stochastic, have broad applications in science and engineering, ranging from climate sciences (Cannarsa et al. 2020, Fisher al.] (200). finance (Pascuccil, 2011). to atomistic simulations Mardt et al. 2018 McCarty
Parrinello 2017, Schütte al. 2001). and open quantum system dynamics (Grini and Kossakowski 1976; Lindblad 1976). among others. However, it is usually the case that no analytical models of the dynamics are available and one must resort to data-driven techniques to characterize dynamical system. Two powerful paradigms have emerged: deep neural networks (DNN) and kernel methods. The latter are backed up by solid statistical guarantees (Steinwart and Christmann, 2008) determining when linearly parameterized models can be learned efficiently. Yet, selecting appropriate kernel function may be hard task, requiring a significant amount of experimentation and expertise. In comparison, the former very effective learning complex data representations (Goodfellow et al., 2216)), and benefit from a solid ecosystem of software tools, making the learning process feasible on large-scale systems. However, their statistical analysis is still in its infancy, with only a few proven results on their generalization properties.
Kernel-base based methods hinge on the powerful idea of characterizing dynamical systems by lifting their definition over a Hilbert space of functions and then studying the associated transfer operators. They describe the average evolution of functions of the state (observables) over time and for deterministic systems are also known as Koopman operators. Furthermore, transfer operators are linear, and under additional assumptions admit a spectral decomposition, which provides a valuable tool to interpret
and analyze the behavior of non-linear systems (see e.g. BBrnnon al. 20222 Kutz al. 2016 and
"qqual contribution, corresponding authors.

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Data Trajectory AAAABAAA
Representation Linear- Nonlinear Kernel
Operator Regression [Kostic et al. 2022, Klus et 2020)
Spectral Decomposition
VAMPNets- - DPNets
Forecasting
Figure Pipeline for learning dynamical systems. DPNets learn data representation to be used with standard operator regression methods. In turn, these are employed to solve downstream tasks such as forecasting and interpreting dynamical systems via spectral decomposition.
references therein). The usefulness of this approach critically relics on an appropriate choice of the observable space. In particular, in order to fully benefit from the spectral decomposition. it is crucial find observable space F which linearizes the dynamics and is invariant under the action of the transfer operator. kernel-based methods, such a space is implicitly linked to the kernel function and gives a clear mathematical meaning to the problem of choosing "good" kernel. Unfortunately, the analytical form of transfer operators is often intractable or unavailable, especially in complex poorly understood systems, posing challenges in constructing invariant representations.
In this paper, we build synergistically upon both kernel and DNN paradigms: we first employ DNNs to learn an invariant representation that fully captures the system dynamics, and then forward this representation to sernel-based algorithms for the actual transfer operator regression task. This general framework is illustrated in Fig. | Our method, named Deep Projection Networks (DPNets), addressing the challenge of providing good representations to the operator regression algorithms, can be cast as an optimization problem over neural networks and can benefit from differentiable and numerically well-conditioned score functional, enhancing the stability of the training process.
Previous work Extensive research has been conducted on learning dynamical systems from data. The monographs (Brunton al. 2022 Kutz al. 2016) are standard references in this field. To Iearn transfer operators we mention the works (Alexander and Giannakis] 2020 [Bovvric and Hamzi] 2017
Das and Giannakis) 2020: Kawahara 2016: Klus et al. 2019:] Kostic et al. [022 Williams et al. 2015b) presenting kernel- based algorithms, and (Azencot el al. 2020 Bevanda et al.
2021 Fan
2021; Lusch et al.) 22118) Morton et al.] 2018) based on deep learning schemes. Finding meaningful representations of the state of the system to be used in conjunction with transfer operator learning is critical challenge, tackled by many authors. We mention the works (Azencot al. 2020 Federici
ct al. 2024 Lusch et al.) 2018: Morton et al. [2018] Otto and Rowley| (2019) where a representation is
learned via DNN schemes, as well as (Kawahara 2016 Li et al. 2017 Mardt et al.) 22199] 2018 Tan
and Wu 2027 Yeung al. 2019 Wu and Noë 2019 addressing the problem of learning invariant subspaces of the transfer operator. Mostly related to our methodology is (Andrew et al.] (2013), which introduced deep canonical correlation analysis and, especially, VAMPnets (Mardt et al., 2018 Wu and Noē] 2019), which repurposed this approach to learn transfer operators. Within our theoretical framework, indeed, we will show that VAMPNets can be recovered as a special case in the setting of discrete dynamics. More in general, operator learning with DNN approaches is reviewed e.g. in Kovachki al. (202)), with a specific facus on PDEs.
Contributions Our contributions are: 1) Leveraging recent results in statistical learning theory, we formalize the problem of representation learning for dynamical systems and design a score function based the orthogonal projections of the transfer operator in data spaces (Sec. D 2) We show how to reliably solve the corresponding optimization problem. optimal representation learning under the mild assumption of compactness of the transfer operator (Thm. D. In the continuous case, our method applies to time reversal invariant dynamics (Thm. 2 including the important case Langevin dynamics; 3) We show how the learned representation can be used within the framework of operator regression in a variety of settings. Numerical experiments illustrate the versatility and competitive performance of our approach against several baselines.
the discrete case, our score leads to
Notation We let N be the set of natural numbers and No- = {0) U N. For m N we denote [m] := {l...., , m). If T is compact linear operator on two Hilbert spaces we let (T) be its i-th largest singular value, we let (T)" be the
pseudo-inverse, inverse, and T the adjoint operator. We denote by || and || [HS the operator and Hilleer-Schmiit norm, respectively.
REPRESENTATION LEARNING FOR DYNAMICAL SYSTEMS
In this section, we give some background on transfer operators for dynamical systems, discuss key challenges in learning them from data, and propose our representation learning approach.

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Background Let (XT) be stochastic process taking values in some state space X. where the time index L can be either discrete (T = No) or continuous (T= +oo)), and denote by let the law of X,- Let F C R* be a prescribed space of real-valued functions, henceforth referred to as
observables space. By letting s, E T. with I t. the forward transfer operator Te.t: F evolves an observable f X + R from time s E T to time t. by the conditional expectation
[T,,())(z) :- (X) [X, = x]. I E X.
(1)
For a large class of stochastic dynamical systems, these linear operators time- tome-homogeneoss that they only depend on the time difference s. this case T.s = To,t- =: Te-s. Further, we will use the shorthand notation T := Ti. Time-homogencous transfer operators at different times are related through the Chapman Kolmogorov equation AAllen 2007). T TT Ta. implying that the family (Tz forms a semigroup. The utility of transfer operators is that, when the space is suitably chosen, they linearize the process. Two key requirements are that F is invariant under the action of Tr, that is [( C F for all t (a property that tacitly assumed above), and rich enough to represent the flow of the process. Two common choices for F that fulfill both requirements are the space of bounded Beasa-mea functions and the space of with respect the invariant distribution π,, when exists. To ease presentation we first focus the latter case, and then extend our results to non- stationary time- homogeneous processes. Formally, the process (Xe) keT admits an invariant distribution π when = a implies 14 = т for all s < (Ross 1995). This in turn allows one to define the transfer operator on F == (X).
functions 1.3 (2t
While in the discrete-time setting the whole process can be studied only through T TI, when time continuous the process is characterized by the infinitesimal generator of the semigroup (Ti)220>
defined as
C == lim (Tx l)/t.
(2)
t-0+
C can be properly defined on the Sobolev space square-integrable gradients (see Lasota and Mackey
(2)) c (2) formed by functions with
47 1994 Ross 1995).
Learning transfer operators In practice, dynamical systems are only observed, and neither T nor its domain F= (X)) are known, providing a key challenge to learn them from data. The most popular algorithms (Brunton et al. 2022, Kutz et al. 2016) aim to learn the action of T: F + F on a predefined Reproducing Kernel Hilbert Space (RKHS) H. forming subset of functions in F. This allows one, via the kernel trick, to formulate the problem of learning the restriction T to H. The H + FF via empirical risk minimization (Kostic et al. (022). However, recent theoretical
advances (Korda and Mezic 2017 Klus et al., 2 et al. 2023), proved that such algorithms are statistically consistent only to The where P' is the orthogonal projection onto the closure H. in F. The projection Ps constrains the evolved observables back inside H. thereby. in general, altering the dynamics of the system. Therefore, to assure that one properly learns the dynamics, two major requirements on H are needed: i) TH needs to approximate well T, i.c. the space H needs to be big enough relative to the domain of T: ii) the difference between the projected restriction and the true one, i.e. the approximation error || |xx]Trx || needs to be small.
When 7 is an eet al. 2022), i.e. H. is dense in F and the approximation error is zero, leading to an arbitrarily good approximation of dynamics with enough data. Still, another key issue arises: the norms the a-priori chosen H and the unknown do not coincide, since the latter depends on the process itself. This metric distortion phenomenon has been recently identified as the source of spurious estimation of the spectra of T (Kostic et al., 2023). limiting the utility of the learned transfer operators. Indeed, even if T is self-adjoint, that is the eigenfunctions are orthogonal in F, the estimated ones will not be orthogonal in H. giving rise to spectral pollution (Kato] 1976). This motivates one to additionally require that iii) H. is a subspace of J, i.e. both spaces have the same norm.
universal RKHS, both the above requirements are satisfied (Kostic
To summarize, the desired optimal H is the leading invariant subspace of T, that is the subspace corresponding to the largest (in magnitude) eigenvalues T. This subspace H. achieves zero approximation error, eliminates metric distortion and best approximates (in the dynamical system sense) the operator T. Since any RKHS H is entirely described by a feature map, learning leading invariant subspace H from data is, fundamentally, a representation learning problem.
Approach We start by formalizing the problem of learning a good finite dimensional representation space for T. and then address the same for the generator L. We keep the discussion less formal here, and state our main results more precisely in the next section. Our approach is inspired by the

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following upper and lower bounds on the approximation error, a direct consequence of the norm change from H to F.
- Px|PP X (Cx) - - I 111 [[11-Pa] Pu]TPPL Xmmll">umax AAmaa(CCu))
(3)
where Cn is the covariance operator on P/ w.r.t the measure n, while X and Amax are the smallest and largest non-null eigenvalues, respectively. Note that the norms on the hypothetical domain H and true domain L (A') coincide if and only if CH = I, in which case equalities hold in B and the
approximation error becomes |||/ Px|T Pxll-
When the operator T is known, the latter quantity can be directly minimized by standard numerical algorithms for spectral computation to find invariant subspaces (see e.g. Golub and Van Loan 2013). Unfortunately, in stochastic setting T is unknown since we cannot compute the conditional expectation (I). To overcome this issue we propose learning approach to recover the invariant space H. which is rooted in the singular value decomposition, holding under the mild assumption that T is compact operato The main idea is that the subspace made of the leading left singular functions of T serves as a good approximation of the desired leading invariant subspace of Namely, due to the orthonormality of the singular functions, we have that C2 = and Px becomes the r-truncated SVD of T. that is, its best rank-r approximation. Therefore, according to B. the approximation error is at most (T)) which can be made arbitrarily small by rising r. Moreover. the distance of the subspace of left singular functions to the desired leading invariant subspace is determined by the "normality" of T (Trefethen and Embree] 2020). If the operator T is normal, that TT TTT, then both left and right singular spaces are invariant subspaces T,, resulting zero approximation error irreppectivly of r. This leads us to the following optimization problem
H. {11 P TPe lis
C# CH dimi r; dim(H') sr
max
(4)
Relying on application of Eckart- desired representation space H can be computed by solving (4). Note that, in general, the auxiliary space Ht' is needed to capture right singular functions, while if we have prior knowledge that T is normal without loss of generality one can set H 71'
Theorem Kato 1976). we can show that the
The same reasoning cannot be applied straight away to the generator a continuous dynamical system, which in is not even guaranteed to be bounded (Lasota and Mackey 1994). let alone compact. For time-reversal invariant processes. however, T and L are self-adjoint, that is 7 T and L = L* This includes the important case of Langevin dynamics, which are of paramount importance for molecular dynamics (see, e.g., Tuckerman, 2010). For such systems we show (see Theorem P that the leading r-dimensional subspace of the generator L can be found by solving the following optimization problem featuring the partial trace objective
max ftr (PM C PH) CC I, dim sr).
(5)
(x))
where now the projector is Pi:: WL W..((X'. We stress that, since we assume L L" we can relax the above assumption on the compactness of the transfer operators (T))2o, and still show that the leading invariant subspace of L is the optimal H in (5). So, on such H we can estimate
well C via generator operator regression, c.g. (Hou al. 2023 Klus et al. 2020).
3 DEEP PROJECTION SCORE FUNCTIONALS
In this section show how solve the representation learning problems 4 and (5) using DNNs. In the discrete case, we consider a generalized version of problem p which encompasses non-stationary processes, for which the probability distributions change along the trajectory. Namely, let X and X' be two X'vuuued random variables with probability measures μ and μ², respectively. Because of hime-homogeneity, w.L.o.g. X models the state at time 0 and X' its evolution after some time f. Then, the transfer operator can be defined on the data-dependent domains Te: L', (X) + (X)). Replacing PH: with Pk (X) + L (X') in 4., and using the covariances CH and CH w.r.t.
'This property is fulfilled by a large class of Markov processes (see e.g. Kostic al. 2022 and is weaker than requiring the operator being Fillher-SShhiie Mardt al. 2018)

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the measures jl and μ², we obtain the appropriate representation learning problem for -tationnny process; see App. B for more details. Within this general setting we optimize two feature maps
Vi : x' R", and Vin :X X
(6)
parameterized by w taking values in some set W. Next, defining the RKHS Hi F
span( Wwws)sefe and H' == span( ( pe)ttr both equipped with the standard inner product, we aim to solve (4) by optimizing over the weights w. To avoid solving the constrained optimization problem, further propose relax the hard constraints in 4 through metric distortion loss R: R'xr_ - R. that zero if and only if it is applied to the identity matrix. Our choice in Section well defined for SPD matrices C, is
R(C) -C -InnC))).
Finally, in Lem. App. we show that can solved by maximizing, for y >
(w()== = (C XCx,(())) (CT,) - (R(CX) +((C))).
(7)
where we introduced the uncentered covariance matrices
(8) Notice that if = 0 and the covariances Cx and Cx are nonsingular, then the score reduces to
OX = E E/ (X)e,, C% == E (X")). and CXx == E oh (X) (X')
Xx X 3 HS which is called VAMP-2 score in the VAMPNets approach of (arrd ct al. 2018 Moreover, in ( the HS norm is replaced by the nuclear norm, the score becomes the objective of canonical correlation analysis (CCA) (Harold 1936 Hardoon et al. 2004) in feature space. When DNNs are used to parametrize (6) such score is the objective of Deep-CCA (Andrew al. 2013), known as VAMP-1 score (Mardt al. 2018) in the context of molecular kinetics. Therefore, by maximizing pn we look for the strongest linear correlation between Ha. and Hig We stress that differently from Deep-CCA and VAMPNets the crucial addition of R is to decorrelate the features within each space. guiding the maximization towards a solution in which the norms of H and Ht' coincide with those of (X) and (X)), overcoming metric distortion.
[KCC))
While in the optimal representation the covariances CX and Cx are the identity, in general they non- non-invertibe invertible, making the score Pp' non- difffeferentablle during optimization. Indeed, unless the rank of both covariances is stable for every w E W. we might have exploding gradients (Golub and Pereyra 1973). Even ignoring differentiability issues, the use of the pseudo-inverse, as well as the use of the inverse in the non- sinnula case, can introduce severe numerical instabilities when estimating pP and its gradients during the training process. More precisely, the numerical conditioning evaluating Pl using 7 is determined by the covariance condition numbers x (C%) (C%) AA, and
Xi (C
(xx) // (Cxx that can be very large in practical situations (see e.g. the fluid dynamics example See. To overcome this issue, we introduce the relaxed score
(Tww)= = ||xxxx Iis/ (|CC || | |) - (R(CF) R(C)))).
(9)
which, as implied by Theorem beloo. is lower bound for the score in 7 A key advantage of this approach is that the score S is both differentiable (apart from the trivial case CX. Cx 0) and has stable gradients, since we avoid matrix inversion. Indeed, computing S is always well-conditioned, as the numerical operator norm has conditioning equal to one.
The following theorem provides the theoretical backbone of our representation learning approach. Theorem T: (x) + (2)( is compact, Hew S (X) and Hi, S (2), then for all 7 220 0
+... (ww) & (w)) S +.. o?
(10)
Moreover, if (Ww.))ee(e) and (Ww.))ee are the leading r left and right singular functions of T. respectively, then both equalities in 40 hold. Finally, if the operator T is Schhert-Schmidt, a a,(T)
(T) and y > 0, then the "only if relation is satisfied up to unitary equivalence.
This result establishes that the relaxed score is tight lower bound, in the optimization sense, for More precisely, in general [S*(w) (w))p"(ww) ] is arbitrarily small for small enough R( R(w), while the feasible set in both scores coincide. For instance, for deterministic linear system this happens as
Here, W and Vin are the j-th component of the and yh. respectively.

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Algorithm DPNets Training
Input: Data D; metric loss parameter y; DNNs tie V/ X' + R,; optimizer U; of steps k.
I: Initialize DNN weights to w1.
2: for j to & do:
Ox, Cx. Cxx + Covariances for Pwy and Vmy from mini-hatch of m S n samples.
3:
4: F(wy)
S- (w;) from 2 and the covariances in Step 3.
5: Wi++1
Umjg, ((rr)) where VF(w) computed using backpropagation.
6:
end for
7:
return representations viux- Vaxe
soon as the classes of features th. and eyy include linear functions. In general, appeal to the universal approximation properties DNN (Cybenko 1989) to fulfill the hypotheses Theorem ||
Generator learning We now consider how to learn the representation of continuous time- homogeneous processes arising from stochastic differential equations (SDE) of the form
dX.- A(X,) dt B(X,) dW.,
(11)
where A and B the drift and diffusion terms, respectively, and W; is Wiener process. The solution
of (IT) is a stochastic process that is time-reversal invariant w.r.t. its invariant distribution x, AAmold
1974). To show that solving ( leads to learning the leading invariant subspace of L. we restrict to
the C L and consider the partial trace of L w.r.t. to the subspace H.. = span span(t)w.)) ) W1.2 (X) as our objective function. Being L self-adjoint, we relax the compactness assumption
of Thm. [ requiring only the much weaker condition that has the largest eigenvalues separated from the essential spectrum Kato
1976). In particular; it holds tr (Pw.. L Pw..) = tr ((Cx) C%)..
where C = [X)=((X) duw (X) is the continuous version of the gross-covariance and diw (-)
given by the Itō formula (see e.g. Arnold 1974 Klus et al. 2020; Hou et al. 2023))
(x) Ma (x) dx/dt + x (dX/dt)
Pe tm (X)d//d) X = x]
(12)
(x)) A(X) + y ((((ss) T v2 (u))(())(t))
(13)
The following result, the proof which is given in App. then justifies our continuous DPNets.
Theorem 2. If H. WL)((X',, and X (C) 2
(C) are eigenvalues of L above its
essential spectrum, then for every > 2 0 holds
()) == ((Cx) (C RR(C)) S (CC +,(C),
(14)
and the equality achieved when ww.j is the eigenfunction of C corresponding to (or j [r].
Notice how tr ((C)) ()) in Eq. 14b is the sum of the (finite) eigenvalues of the symmetric
eigenvalue problem C.v, XC Thus, con-pathological cases, its value and gradient can be
computed efficiently in a numerically stable way, see e.g. (Andrew and Tan 1998).
4 METHODS
Learning the representation To optimize a DPNet representation one needs to replace the population covariances and cross-covariance in Eqs 7 and (14) with their empirical counterparts. In practice, given tin. Vy. X + RR' and a dataset D (Ix, d))eem) consisting of samples from the joint distribution of (X) one estimates the needed covariances C% Cx and CXxx
CX Σ (w (x:) Cx- FE(M)
He (c4)% and Xxx #E(m]
Σ
Σ (c())) VeIm
In App. D.1 we prove that the empirical scores P7: W IR and Sy W R, that is the scores computed using empirical covariances, concentrate around the ones with high probability for any fixed w. Obtaining uniform guarantees over w requires advanced tools from empirical processes and regularity assumptions on the representations which could be the subject of future work. The difficulties arising from the estimation of PY during optimization are also addressed.

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Alg. we report the training procedure for DPNets-relaxed (using score ). which can also be
applied to 7 and 14 suitably modifying step 4. When using mini-batch SGD methods to compute
VP m (w) or V J. the batch size m should be taken sufficiently large to mitigate the impact of biased stochastic estimators. The time complexity of the training algorithm is addressed in App. D.4
Operator regression Once a representation the is learned, it can be used within the framework of operator regression, see Fig. | Following the reasoning in (Kostic et al. 2022) one sees that any model Twu H. + Hw of the transfer operator T acts on functions in Hu. as
h2 == v.)
Twh := :-v T2, R",
(15)
where 7 E Rrrr is a matrix in the representation space. For example, denoting the data matrices Ve = (x1) ... (xn)),,, V = (x1) ... 1o E IRrx", the ordinary least square estimator (OLS) minimizes the empirical risk 1 T Wullins yielding 7= ( ( WT.
Downstream tasks Once i is obtained, it can be used predict the next expected state of the dynamics given the initial one, compute the modal decomposition of an observable (Kutz et (0T6), estimate the spectrum of the transfer operator (Kostic et al.] 2023) or controlling the dynamics (Proctor et al.) 2016). Indeed, recalling that X' is the one step ahead evolution of X, we can use (15) to approximate (X') X = x| as v (x) [T.. Moreover, relying on the reproducing property of Hw. the predictions can be extended to functions f: X + R' as [X"( X x] ~ (z) ( It
where F = [f(z)] f(2,, [(,,)]= E R is the matrix of the observations of the evolved data. Clearly, when the observable is the state itself (i.e. f(x = x),, we obtain one step ahead predictions.
Next, observe that eigenvalue decomposition of the matrix T leads to the eigenvalue decomposition of the operator T. Namely, let (A,, ū, 8.) E Cx x Cd be an eigen- triplet made of eigenvalue, left eigenvector and right eigenvector of T, that is T7, A, प. λقū and iz 0j. dg i,k [r], we directly obtain, using (5). the spectral decomposition
Te ExELy J. 9s.
where f(x)) = (=) and (x(x) (T.) (w)
(16)
Finally, we can use the spectral decomposition of 7 to efficiently forecast observables for several time-steps in the future via E/,, (X., X, x] = Eneny X (0.. f((z). which is known as the extended dynamic mode decomposition (EDMD), see e.g. (Brunton et al. 2022). Noticing that
(g. bz) = (ū.)* ² = TT= As when 2U * O. and, again, using the reproducing property, we can extend this approximation to veltor observables f: X +++12 R2 as
(Xx( Xu x] = EvEry (v) (RR)
(17)
5 EXPERIMENTS
In this experimental section we show that DPNets (i) learn reliable representations of non-normas dynamical systems, (ii) can be used in its continuous-dynamic form (14) to approximate the eigen- values of Langevin generator, and (iii) outperforms kernel methods and auto-encoders in large-scale and/or structured (c.g. graphs and images) data settings. To have a fair comparison, every neural network model in these experiments has been trained on the same data splits, batch sizes, number of epochs, architectures and seeds. The learning rate, however, has been optimized for each one separately. We defer every technical detail, as well as additional results to App. PP The code reproduce the examples can be found at http://ppppppppt.tolllll..ccl gith 10/ DPNets/ and it heavily depends on Kooplearn
Baselines We compared methods DPNets (P*) and DPNets-relaxed (S*), where appropriate. with the following baselines: Dynamic AE (DAF) of (Lusch et al. 2018), Consistent AE of (Azencot
et al,, 2020), DMD of (Schmid, (2010). KernelDMD and ExtendedDMD of (Williams et al. 201 b). and VAMPNets of (Mardt et al. 20088)
Logistic Map We study the dynamical system Xe41 (4X, X,) + &.) mod 1, for X' [0, 1) and & being i.i.d. trigonometric noise (Ostruszka et al. 2000) The associated transfer operator is non-normal, making the learning of its spectral decomposition particularly challenging (Kostic et al.
2023). Since T can be computed exactly, we can sidestep the problem of operator regression (see

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Representation
Spectral Erior
Optimality Gap
Model
P
Transition
Enthalpy AH
DPNets
0.28 (0.03)
0.64 (0.01)
DPNets
12.84
17.59 ns
-1.97 keal/mol
DPNets-relaxed
0.06 (0.05)
1.19 (0.04)
Nys-PCR
7.02
5.27 ns
-1.76 kcal/mol
VAMPNets
0.21 (0.09)
0.97 (0.22)
Nys-RRR
2.22
0.89 ns
-1.44 kcal/mol
Cheby-T T
0.20
1.24
NoiseKernel 0.19
2.17
Reference
40 ns
-6.1 kcal/mol
Table I: Logistic Map. Comparison of DPNets to
Table 2: Chignolin. Comparison between DP- Nets and kernel methods with Nyström sampling (Meanti et al. )023
relevant baselines. Mean and standard deviation are over 20 independent runs. We used r = 7.
Fig. D. and focus directly on evaluating the quality of the representation encoded by H... We evaluate DPNets on two different metrics: (i) the optimality gap Di=1 of( (w)) for r = 3 and (ii) the spectral error, given by max, miny } (1xx Tix) (T).. While (i) informs on how close one is to solve (ii) measures how well the true eigenvalues of can be recovered within the representation space Hww In Tab. | we compare the DPNets representation against VAMPNets, EXtendedDMD with a feature map of Chebyshev polynomials and the feature defining the trigonometric noise OOtruszka al. (2000) of the process itself. Notice that in this challenging setting recovering eigenvalues via singular vectors is generally problematic, so a moderate optimality gap may lead to a larger spectral enror. In Fig. 2 we report the evolution of the spectral error during training for DPNets and VAMPNets, while we defer App. F. an in-depth analysis of the role of the feature dimension. Notice that DPNets and APNets-relaxee excel in both metrics.
Continuous dynamics We investigate one oie-dimenioonal continuous SDE describing the stochastic motion of particle into the Schwantes potential (Schwantes and Pande 2013). The invariant distribution for this process is the Boltzmann distribution x (dz) xx e (-3V(F) dr. where V(x) is the potential at state x. The non-null eigenvalues of L' hold physical significance, their absolute value represents the average rate at which particles cross one of the system's 's potential barriers (Kramers) 1940): our objective is to accurately estimate them. Here X, is sampled non- con-uuiformly according to a geometric law. In the lower panel of Fig. 2 we report the estimated transition rates, of L along the DPNets training loop. Notice that the embedding the progressively improves eigenvalue estimation, indicating that the invariant subspaces of the generator C are well captured.
Ordered MNIST Following Kostic et al. (2022), we create stochastic dynamical system by randomly sampling images from the MNIST datasel according to the rule that X, should be an image of the digit t (mod 5) for all t E Ng. Given an image from the dataset with label c, a model for the transfer operator T of this system should then be able to produce an MNIST-alike image of the next digit in the cycle. In the upper panel of Fig. 3 we thus evaluate DPNets and number of baselines by how accurate is an "oracle" supervised MNIST classifier (test accuracy for in-distribution 2 99%) in predicting the correct label c t (mod 5) after steps of evolution. DPNets consistently retain accuracy above 95%, while for every other method it degrades. The "reference" line corresponds to random guessing, while the "Oracle-Feature" baseline is an operator regression model (EDMD) using, as the dictionary of functions, the output logits of the oracle, and despite having been trained with the true labels, its performance degrades drastically after 2 5.
Fluid dynamics We study the classical problem of the transport of a passive scalar field by 2D fluid flow past a cylinder (Raissi et al. 2020). Each data point comprises regular 2D grid that encompasses fluid variables, including velocity, pressure, and scalar field concentration at each grid point. This system is non-stationary and is also known to exhibit non-normal dynamics (see Trefethen and Embree 2020). We evaluate each trained baseline by feeding to it the last snapshot of the train trajectory and evaluating the relative RMSE (that is, the RMSE normalized by the data variance) between the forecasts and the subsequent test snapshots. In this experiment, we also have a physics-informen (PINN) baseline not related to transfer operator learning, which is however the model for which this dataset was created in the first place. Remarkably, the forecasting error of DPNets does not grow sensibly with time as it does for every other method.
The Metastable states of Chignolin In our last experiment, we study the dynamics of Chignolin, a folding protein. from 106 ps long molecular dynamics simulation sampled every 200 ps. totalling over 500, 000 data points (Lindorff-Larsee Larsen et al. (01T). We focus on the leading eigenfunctions of the transfer operator, which are known (Schütte et al. (00T) to provide simple characterization of the slowest (and usually most important) physical processes occurring along the dynamics. From the leading left eigenfunctions of T, one can indeed construct the free energy surface (see Fig.

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Reference
DPNets - relaxed
DynamicalME
DMD
Featurss
DPNets
VAMPNets
ConsistentAE
KemelDMD
PINNs
Lggistc map spectral error
Ordered MNIST Accuracy
DPNets-relaxed
0.20
1.0
0.15
0.5
0.05
Nyström KernelDMD
Langevin's transitions rate
Fluid Flow relative RMSE
10a
10a
10a2
10a2
10a
10-3
Training Steps (k)
Prediction Steps
Free Energy Surface
Eigenvalue error decays during training. Up- per: spectral eтror for the lo- gistic map. Lower: DPNets- estimated eigenvalues of C for the Langevin dynamics.
Figure 2:
Figure 3: Forecasting with
Figure 4: Free energy sur- face of the 2 slowest modes Chignolin,estimated by DP- Nets and Nystfom PCR. To be compared with Bonati et al. 2021
DPNets. Upper: classification accuracy over time for ordered MNIST. Lower: forecasting RMSE over time fluid dy- namic example.
whose local minima identify the metastable states of the system. The free energy surface, indeed, is proportional the negative log-pdf of finding the system given state while thermodynamic equilibrium, meaning that a state of low free energy is highly probable, hence metastable. For Chignolin, the metastable states have been thoroughly studied (see e.g. Novelli et al., 2022). The
leading left eigenfunction of T encodes the folding-unfolding transition and the typical transition time the implied timescale (Mardt et al. 2019) associated with its eigenvalue. The difference between the local minima the free energy surface encodes the enthalpy AH of the transition. In Tab. we compare these quantities to the reference values reported in Lindorti- Larsen et al, (201T). We trained GNN-hased DPNet-relaxed, possibly due to the large scale
both DPNets unrelaxed and VAMPNets failed converge, data. compared to KernelDMD estimator trained with the recent Nyström sketching technique (Meanti ct al. (222) as classical kernel methods are completely intractable at this scale system, which match analysis of (Bonati et al. 2021 see App. F Foro more discussion.
D1 Notice that DPNets succeed in finding additional meta-stable states of the
CONCLUSIONS
We propose framework for learning representation of dynamical systems, based on orthogonal projections data-spaces. captures leading invariant subspace the transfer operator and can be applied to both discrete and continuous dynamical systems. In the discrete case, the representation is learned through the optimization of smooth and numerically well conditioned objective function. Extensive numerical experiments demonstrate the effectiveness and generality of DPNets in various settings, suggesting that they are promising tool for data-driven dynamical systems. A limitation this work is that the score functional for the continuous systems might be unstable since it leverages covariance matrix inversion. Moreover, a future direction would be study the statistical learning properties the algorithm presented here.
'A hack-of-theeeenvllpe calculation shows that 450 GBs would be needed just kemee matrices single precision.

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ACKNOWLEDGEMENTS
We acknowledge the financial support from the PNRR MUR Project PE000013 CUP J53C22003010000 "Future Artificial Intelligence Research (FAIR)", funded by the European Union NextGencrattinEEU EU Project ELIAS under grant agreement No. 101220 and the chair "Business Analytic for Future Banking" sponsored by Natixis-Groupe BPCE.
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Reviews:
[{"review_id": "tCKFi69Hum", "reviewer": "Reviewer_hdvQ", "summary": "This paper proposes a method for learning a representation of continuous and discrete stochastic dynamical systems that achieves state of the art results on many datasets.", "questions": "See weaknesses", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper identifies the learning problem as an optimization problem and provides an efficient way to solve it. To overcome singularity, they have a relaxed score function and they also prove that the relaxed score has theoretical guarantees. The proof is mathematically solid.\n\n2. Many experiments are done to verify their claims with impressive results.", "weaknesses": "1. The computation for covariances for $\\psi_{w_j}$ and $\\psi'_{w_j}$ might be costly. How does this computation cost compare to the previous methods? Is there a way to speed up?\n\n2. For the continuous-time dynamic, the score function might be unstable. Have you observed this in experiments?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "PEyRpt53FV", "reviewer": "Reviewer_sgxX", "summary": "This paper presents a relatively new approach to learning the operator for dynamical systems. Instead of one-step modeling of the neural operator, the authors suggest first learning the invariant representations, and then leveraging these learned representations to perform operator regression. The authors assessed their results on various datasets, including an ordered MNIST, a 2D fluid flow, and a molecular dynamics simulation, primarily using metrics such as spectral error and RMSE.", "questions": "Q1. I am curious about the empirical performance of the results concerning different hyperparameters, particularly the value of $r$.\n\nQ2. Additionally, what would be the loss for the OLS estimator of the transfer operator across different datasets? It would be concerning if the performance is highly sensitive to the value of $r$.\n\nQ3. For the dynamical data, i.e., the 2D fluid flow and the molecular dynamics, it would be beneficial if the authors could provide the visualizations of the dynamics.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This paper provides a new way of learning invariant representations of dynamical system.\nThe authors provide theoretical justifications for the derivation of their objectives functions.", "weaknesses": "W1. The pipeline design suggests that there might be a possibility of error accumulation from the first phase of learning representations to the second phase of learning dynamics. It would be beneficial if the authors could provide empirical justification to show that their design is comparable to or better than learning the neural operator in a single step. For instance, establishing baselines using Frameworks like FNO or Deeponet would be ideal.\n\nW2. Tying in with W1, the efficacy of the proposed method, particularly the operator regression in the second part, remains uncertain when applied to chaotic systems where acquiring accurate representations in the initial phase may prove challenging\n\nW3. It would be great if the authors could evaluate the performance on insightful metrics like Lyapunov spectrum. And including discussion on recent related works (e.g., https://arxiv.org/pdf/2304.12865.pdf) should be necessary.\n\nW4. If the primary goal is representation learning, I feel that discussions and comparisons related to popular representation learning methods (e.g., contrastive learning) should be discussed and included.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "rBO3BikRlF", "reviewer": "Reviewer_1QLL", "summary": "In this paper a particular representation of a state space of a dynamical system is learned. \nThe representation is intended to correspond to the subspace of highest singular values of the transfer operator of the system, of a fixed predifined dimension. It is learned from the observed trajectories of the system and parametrised by neural networks. \n\nThe corresponding optimisation objective can be expressed in terms of the covariance matrices of the data (in the representation space).  While the direct involves matrix inversion and is computationally difficult, a proxy objective, which is proven to be a lower bound, is proposed. The proxy objective has better computational properties. \n\nOnce the representation is learned, the transfer operator can be learned using standard approaches based on operator regression.  Experiments are performed to demonstarte a competitive performance of the approach.", "questions": "In addition to the points above:\n\nIs the relaxation objective (9) a new contribution of this paper, or were similar relaxations employed in previous work (for instance VAMPSNets, DeepCCA)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper is clearly written and the topic of modeling dynamical systems is important. \nThe proposed representation space is very natural.", "weaknesses": "This is mainly an empirical paper, as it proposes to model a certain space using neural networks with gradient descent optimsed objective.  However, the empirical evaluation of the method is not completely convincing, and some points are not clear.  \n\n\nIn addition, even the relaxed objective introduced in the paper is computationally heavy. It involves \ncomputing the covariannce matrices as _differentiable functions_. This means the computation graph involves summing $n$ matrices of size $r \\times r$, each coordinate of which is a differnetiable function. \nAs noted in the paper, minibatches could be used but would introduce bias. Even with minibatches, the \nrepresentation dimension $r$ would have to stay low, and appears to be low in all experiments (in which it was specified). \n\n\nA more detailed notes on experiments:\n\n**Ordered MNIST:** My understanding is that this example is new in this paper. Is there an explanation/intuition why other methods fails here? Are convolutional networks used to model the \nrespresentation? I assume the non NN methods fail simply since representing images is easier with convolutions. Why do other methods fail? I believe explaining the difference here could make the paper considerably stronger. \n\n\n**Logistic Map:** The takeaway here is not clear.  DPNets has the worst spectral error among all methods. \nIn addition, there appears to be no theoretical reason why DPNets-realxed should perform better on this metric, as it was introduced as a computational relaxation. Do the suggest it has properties allowing it to be a better estimator in general? \n\n**Continuous dynamics:**\nThe scale of Fig. 2 lower pane is that of 200. This makes is very hard to understand whether the smaller \neigenvalues are estimated accurately and what is the order of magnitude.  Displaying ratios estimated/true would be more informative. Also, are there other ways to estimate the eigenvalues? \n\n\n**FlowRMSE:**\nThe general magnitude of the quantities in this experiment is not clear.  If the quantities are of order 10, then the difference between errors of order 0.1 and 0.001 is less impactful, and all methods perform well. \nWhat are the typical values of the quantities for which RMSE is computed? Again, percentage ratios would be much clearer than absolute values. \n\n\n\n**Chignolin Experiment:**\n In Table 2, describing the Chignolin experiment, while estimated quantities are somewhat closer to the reference values than the true quantities, they still look quite far. It is thus not clear what is the point of this part of the experiment.  \nThe second part of that experiment, involving Fig.4, unfortunately can not be understood from the main text. (what is the free energy surface, what is known about its proper minima, why matching the results of Bonati et al   is of interest? ).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "lbHV6dGFeK", "reviewer": "Reviewer_UfpG", "summary": "The authors provide a method to learn an invariant representation for dynamical systems helps give an approximation of the transfer operator. They do this while also alleviating weaknesses of previous approaches, such as metric distortion. To do this, the authors show that the search for a good representation can be cast as an optimization problem over the space of neural networks. They provide and an objective function, and tractable approximations to it. They compare their method to other state-of-the-art approaches.", "questions": "**Questions:**\n- It seems in most experiments, the output dimension of the neural networks, $r$, is low. Would this be the case for most practical applications?\n- On page 7 in \"Downstream tasks\", in order to make predictions of the next step, the authors need to compute: $(\\hat{\\Psi}_{w}^\\top)^\\dagger$, i.e. the pseudo-inverse, is that right? If so, it seems this would slow down the computation of the predicted next steps, so how do the computation times compare with other methods for predicting next steps? Any numerical instabilities?\n- In Algorithm 1, the authors use $\\hat{\\mathcal{S}}^\\gamma_m$, implying this is the relaxed version, is that right? If so, this isn't stated anywhere in Algorithm 1. Perhaps the authors should have two Algorithms that gives the regular DPNets and also DPNets-relaxed.\n- When training the unrelaxed DPNets, how was $\\mathcal{P}^\\gamma$ evaluated? It is stated in page 5 that in general the covariances are non-invertible and thus $\\mathcal{P}^\\gamma$ is non-differentiable during the optimization. Also, as the authors state, the use of the pseudo-inverse can introduce numerical instabilities. How much of a problem is this, in your experience? I see that some experiments were unable to use unrelaxed DPNets because of this.\n- For Figure 3, in the upper panel - \"Ordered MNIST accuracy\", I'd imagine that the \"Reference\" (dashed-line) is just random guessing, i.e. $1/5 = 0.2$? I don't see this stated anywhere, and would like it explicitly stated.\n- For Figure 3 in the lower panel, \"Fluid Flow RMSE\", how many prediction steps were taken? Namely, what are the values of the $x$-axis?\n\n**Remarks:**\n- On page 5, on the line above equation (8), the authors say \"we show in Lem. 3 in App. B\", but I think you mean Section C in the appendix/supplementary materials.\n- On page 6, in \"Learning the representation\", might also need to state that we are also given $\\psi'$, i.e. \" In practice,given a NN architecture $\\psi$ and $\\psi$'. \"", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- The approach seems novel and interesting enough.\n- The authors list the weaknesses of previous approaches (e.g. the weaknesses of previous methods on page 3, under \"Learning transfer operators\") and show how their method attempts to alleviate them.\n- While the authors don't optimize over the main objective, equation (4), they provide tractable approximations, and also qualify and quantify when these approximations are good (e.g. Theorem 1, Lemma 3 in Section C of the supplementary materials).", "weaknesses": "- The organization and the explanation of the method could be more straightforward. For example, Algorithm 1 is rather terse, and a more expanded version would be instructive to the reader. It's hard to get birds-eye view of the method because the explanation is spread throughout the text. Perhaps a separate section in the supplementary materials that gives a more direct explanation of the method would help the reader.\n- The method requires computing covariance matrices of neural network outputs, which could affect the computation if the output dimension is high.\n- I have some reservations about the experiments:\n    - In the beginning paragraph of section \"5 Experiments\", the authors state they only optimize over the learning rate to keep things fair, but I can't seem to find the set of learning rates that were optimized over.\n    - On the previous point, the authors state at the top of page 7 that the mini-batch size should be sufficiently large to mitigate the impact of biased stochastic estimators. But, some neural network models benefit from the mini-batch stochasticity of SGD, as opposed to full-batch normal gradient descent. In this case, I would have liked to have seen optimizing over the batch sizes as well, and not just the learning rate.\n    - I'm also not sure if optimizing over just the learning rate this is completely fair. I would have liked to see optimizing over more hyperparameters, or at least in addition to the current experiments, to also provide the performance evaluation of the original model implemented in the paper, if possible. For example, in the fluid flow example, Azencot et al. (2022) has a similar fluid flow example, but uses a different architecture and I would have liked to see this one in comparison to the author's method.\n    - In the molecular dynamics example, I see Ghorbani et al. (2022) have done similar experiments, but at a smaller scale. Even though the authors performed their experiment at a larger scale, I would have liked to have seen a comparison with Ghorbani et al. (2022) at the smaller scale, or even at the larger scale. This would allow comparing accuracy and training times.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)