Critical Sampling for Robust Evolution Behavior Learning of Unknown Dynamical Systems
Authors: Ce Zhang
Siqi Wu
Kailiang Wu
Zhihai He
Venue: Submitted to ICLR 2023 (2023)
Keywords: Critical Sampling
Evolution Behavior Learning
Dynamical Systems

Abstract: We study the following new and important problem: given an unknown dynamical system, what is the minimum number of samples needed for effective learning of its governing laws and accurate prediction of its future evolution behavior, and how to select these critical samples? In this work, we propose to explore this problem based on a design approach. Specifically, starting from a small initial set of samples, we adaptively discover and collect critical samples to achieve increasingly accurate learning of the system evolution. One central challenge here is that we do not know the network modeling error of the ground-truth system state, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where a forward evolution network and a backward evolution network are designed to learn and predict the temporal evolution behavior in the forward and backward time directions, respectively. Very interestingly, we find that the desired network modeling error is highly correlated with the multi-step reciprocal prediction error. More importantly, this multi-step reciprocal prediction error can be directly computed from the current system state without knowing the ground-truth or data statistics. This allows us to perform dynamic selection of critical samples from regions with high network modeling errors and develop an adaptive sampling-learning method for dynamical systems. To achieve accurate and robust learning from this small set of critical samples, we introduce a joint spatial-temporal evolution network which incorporates spatial dynamics modeling into the temporal evolution prediction for robust learning of the system evolution operator with  few samples. Our extensive experimental results demonstrate that our proposed method is able to dramatically reduce the numbers of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times, especially for high-dimensional dynamical systems.

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CRITICAL SAMPLING FOR ROBUST EVOLUTION BE- HAVIOR LEARNING OF UNKNOWN DYNAMICAL SYS- TEMS
Anonymous authors Paper under double-bling blind review
ABSTRACT
We study the following new and important problem: given an unknown dynamical system, what is the minimum number of samples needed for effective learning of its governing laws and accurate prediction of its future evolution behavior, and how select these critical samples? In this work, we propose to explore this problem based on a design approach. Specifically, starting from a small initial set of sam- ples, we adaptively discover and collect critical samples to achieve increasingly accurate learning of the system evolution. One central challenge here is that we do not know the network modeling error of the ground-truth system state, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where a forward evolution network and a backward evolution network are designed to learn and predict the temporal evo- lution behavior in the forward and backward time directions, respectively. Very interestingly, we find that the desired network modeling error is highly correlated with the multi-step reciprocal prediction error. More importantly, this multi-step reciprocal prediction error can be directly computed from the current system state without knowing the ground-truth or data statistics. This allows us to perform a dynamic selection of critical samples from regions with high network modeling errors and develop an adaptive sampling-learnin To achieve accurate and robust learning from this small set of critical samples, evolution network which incorporates spa- tial dynamics modeling into the temporal evolution prediction for robust learning of the system evolution operator with few samples. Our extensive experimen- results demonstrate that our proposed method is able dramatically reduce the number of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times, especially for high-dimensional dynamical systems.
method for dynamical systems.
we introduce joint spatial-tempont
INTRODUCTION
Recently, learning-bass based methods for complex and dynamic system modeling have become an im- portant area of research in machine learning. The behaviors of dynamical systems in the physical world are governed by their underlying physical laws (Bongard & Lipson] 2007 Schmidt & Lipson 2009). In many areas of science and engineering, ordinary differential equations (ODEs) and partial differential equations (PDEs) play important roles in describing and modeling these physical laws
(Brunton al. 2016, Raissi 2018 Long al. 20188] Chen et al. 2018 Raissi et al. 2019 Qin ct al. 2019). In recent years, data-driven modeling of unknown physical systemss from measure- ment data has emerged as an important area of research. There are two major approaches that have been explored. The first approach typically tries to identify all the potential terms in the unknown governing equations from a priori dictionary, which includes all possible terms that may appear in
the equations (Brunton et al. 2016 Schaeffer & McCalla] 2017 Rudy et al., 2017 Raissi 2018 Long et al. 2018 Wu & Xiu 2019 Wu et al.] 2020] Xu & Zhang 2021). The second approach for
data-driven learning of unknown dynamical systems to approximate the evolution operator of the underlying equations, instead of identifying the terms in the equations (Qin et al. 2019 Wu & Xiu
2020 Qin al. 2021a|
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So Initial Samples
DNN System Model
DNN Model Error
Critical Sampling
Spatial Dynamics
Tw
Multi-Step Reciprocal
Temporal
Jo
S
Evolution
Prediction
Figure 1: Illustration of the proposed method of critical sampling for accurately learning the evolu- tion behaviors of unknown dynamical systems.
Many existing data-driven approaches for learning the evolution operator typically assume the avail- ability of sufficient data, and often require a large set of mcasurement samples to train the neural network, especially for high-dimensional systems. For example, to effectively Iearn neural net- work model for the 2D Damped Pendulum ODE system, existing methods typically need more than 10000 samples achieve sufficient accuracy (Qin et al.) 2019 Wu & Xiu 2020)_ This number increases dramatically with the dimensions of the system. For example, for the 3D Lorenz system, the number of needed samples used in the literature is often increased to one million. We recognize that, in practical dynamical systems, such as ocean, cardiovascular and climate systems, it is very costly to collect observation samples. This leads to a new and important research question: what is the minimum number of samples needed for robust learning of the governing laws of an unknown system and accurate prediction of its future evolution behavior?
Adaptive sample selection for network learning, system modeling and identification has been stud- ied the areas of active learning and optimal experimental design. Methods have been developed for global optimization of experimental sequences (Llamosi et al. 2014), active data sample gener- ation for time-series learning and modeling (Zimmer et al. 2018)) Kriging-based sampling method for learning
dynamics of systems (Huang al.) 2022), adaptive training of physics- informed deep neural networks (Zhang & Shaficezadch 2022). and data-collection scheme for sys- tem identification (Mania et al. 2022p. However, within the context of deep neural network model- ing of unknown dynamical systems, the following key challenging issues have not been adequately addressed: (1) how to characterize and estimate the prediction error of the deep neural networks? (2) Based on this error modeling, how to adaptively select the critical samples and successfully train the deep neural networks from these few samples?
Figure illustrates the proposed method of critical sampling for accurately learning the evolution behaviors of unknown dynamical systems. We start with small set of initial samples, then itera- tively discover and collect critical samples to obtain more accurate network modeling of the system. During critical sampling, the basic rule is to select the samples from regions with high network mod- eling errors so that these selected critical samples can maximally reduce the overall modeling error. However, the major challenge here is that we do not know network modeling error, i.e., the differ- ence between the system state predicted by the network and the ground-truth which is not available for unknown systems. To address this challenge, we establish a multi-step reciprocal prediction framework where a forward evolution network and backward evolution network designed to learn and predict the temporal evolution behavior in the forward and backward time directions, re- spectively. Our hypothesís is that, if the forward and backward prediction models are both accurate, starting from an original state A, if we perform the forward prediction for K times and then perform the backward prediction for another K times, the final prediction result A should match the original state A. The error between A and A is referred to as the multi-step reciprocal prediction error.
Very interestingly, we find that the network modeling error is highly correlated with the multi-step reciprocal prediction error. Note that multi-step reciprocal prediction error can be directly computed from the current system state, without the need to know the ground truth system state. This allows us to perform a dynamic selection of critical samples from regions with high network modeling errors and develop an adaptive learning method for dynamical systems. To effectively learn the system evolution
evolution from this small set of critical samples, we introduce joint spatial-tempon network structure which couples spatial dynamics learning with temporal evolution learning. Our extensive experimental results demonstrate that our proposed method is able to dramatically reduce the number of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems. This paper has significant impacts in practice since collecting samples from real-world dynamical systems can be very costly or limited due to resource/labor constraints experimental accessibility.
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The major contributions of this work can be summarized follows. (1) We have successfully developed a multi-step reciprocal prediction approach to characterize the prediction errors in deep neural network modeling of unknown dynamical systems. We have made an interesting finding that the network modeling error is highly correlated with the multi-step reciprocal prediction error, which enables to develop the critical sampling method. (2) We have designed a joint spatial-temporal evolution network which is able to learn the system evolution with few critical samples. (3) Our proposed method is able dramatically reduce the number of needed samples and related sample collection costs for learning the system evolution, which is highly desirable in practical applications.
RELATED WORK
In this section, we review existing research closely related to our work.
(1) Data-driven modeling of unknown physical systems. There are two major approaches that have been explored in the literature. The first approach aims to learn the mathematical formulas or expressions of the underlying governing equations. In a series developments of this direction, the
seminal work was made by (Bongard & Lipson| 2007 (Schmidt & Lipson] 2009), where symbolic regression was proposed to learn nonlinear dynamic systems from data. Later, more approaches have
been proposed in this direction, including but not limited to sparse regression (Brunton et al.
2016
Schaeffer & McCalla 2017 Rudy et al.] (017)). neural networks (Long et al. 2018 Raissi 2018
Long et al. 2019), Koopman theory (Brunton et al. 2017), Gaussian process regression (Rassi al.
2017 model selection approach (Mangan et al.) 2017). classical polynomial approximations (Wu
& Xiu 2019 Wu et al.) 2020), genetic algorithms (Xu et al. 2020 Xu & Zhang 2021 Xu al.
2021
and linear multi-step methods (Keller & Du 2021). etc.
The second approach aims to approximate the evolution operator of the underlying dynamical sys- tem typically via a deep neural network, which predicts the system state for the next time instance from the current state (Wu & Xiu, 2020) Qin al.) [021a]] [i al., 2021a (b) In fact, the idea of such an approach is essentially equivalent to learning the integral form of the underlying unknown differential equations (Qin et al. 2019). The performance of this approach has been demonstrated for learning ODEs (Qin et al. (019)) and modeling PDEs in generalized Fourier spaces (Wu & Xiu 2020) and physical space (Chen et al. 20022). Recently, this approach has also been extended to data- driven modeling of parametric differential equations (Qin et al. 2021b), non-autonomous systems (Qin et al. 2021a), partially observed systems (Fu et al. 2020). biological models (Su et al. 202T), and model correction (Chen & Xiu 2021 For autonomous dynamical system, its evolution op- erator completely characterizes the system evolution behavior. Researchers have demonstrated that the evolution operator, once successfully learnt, can be called repeatedly to predict the evolution behaviors of the unknown dynamical systems (Qin et (019 Wu & Xiu 2020 Chen et al.) 2022).
(2) State-space models and adjoint state methods. State-space models have shown to be pow- erful tool for modeling the behaviors of dynamical systems (McGoff et al.] (015)). Methods have been developed for approximating dynamical systems with hidden Markov model (HMM) (Rabiner)
1989 Fraser] 2008), recurrent neural network (RNN) (Han et al.) 2004), long short-term memory
network (LSTM) (Vlachas et al. 2018), reservoir computing (RC) (Inubushi & Yoshimura 2017
structured variational autoencoder (SVAE) (Johnson et al. 2016). linear dynamical system (LDS)
and its variations (Fox et al. 2008 Linderman et al. 2017 Gao et al. 2016).
We recognize that the multi-step step forward and backward processes and the usage of mismatch errors are related to those in the recent adjoint state methods for neural ODE learning (e.g. [Cenn et al.] 2018); /Zhuang et al. (2020abb). However, our method is uniquely different in the following as- pects. (1) In the adjoint state method, the back propagation is used to compute gradients based on Lagrangian functional. However, in method, the backward network is used to learn the in- verse of the forward evolution operator (namely, backward evolution operator, see Lemma C.I. (2) The adjoint state method aims to compute the gradients of the loss functions more efficiently and accurately. However, our method aims to discover the critical samples for network learning.
METHOD
In this section, we present our method of critical sampling for accurate learning of the evolution behaviors for unknown dynamical systems.
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3.1 PROBLEM FORMULATION AND METHOD OVERVIEW
In this work, we focus on learning the evolution operator Pa : R" + RR for autonomous dynamical systems, which maps the system state from time | to its next state time ++ A: u(l + A) $ (u((()) - It should be noted that, for autonomous systems, this evolution operator & a remains invariant over time. It only depends on the time difference A. For an autonomous system, its evolution operator completely characterizes the system evolution behavior QQin et al., 2019 Wu & Chen et al. 2022).
Xiu 2020
Our goal is to develop deep neural network method to accurately learn the evolution operator and robustly predict the long-term evolution of the system using a minimum number of selected critical samples. Specifically, to learn the system evolution over time, the measurement samples for training the evolution network are collected in the form of pairs. Each pair represents two solution states along the evolution trajectory at time instances t and + A. For simplicity, assume that the start time is = O. Using a high-accuracy system solver, we generate J system state vectors {u' (0)) at time 0 and {u' (A)())) (A)) at time A in the computational domain D. Thus, the training samples are given by
S = 0 (0) + u² u (0), n² (A) E 1
(1)
is used to train the forward evolution network Fg which approximates the forward evolution operator Ps. As discussed in Section | we introduce the idea of backward evolution operator VA : R" + R", u(0)- = a (u())) The original training samples in Sp can be switched in time
to create the following sample set
S (I) + u (0) J)
(2)
which is used to train the backward evolution network Go. The forward and backward evolution networks, Fe and Go. form a reciprocal prediction loop.
B(O)
RiK- 1) Go
Go 11
B(2) Go
M(K 1)
Go....
F
Jo
((2)
DCK-
a(1)
((2)
Robust Evolution Prediction
2r(2))
M(X 1)
Fo m Forward Evolution Prime Network
Go Backward Evolution
Nelwork
Ground Truth Solution
Inaccurate Evolution Prediction
Figure 2: Illustration of the proposed idea of multi-step reciprocal prediction error.
Based on the forward and backward prediction errors, we introduce a multi-step reciprocal predic- tion process. As illustrated in Figure 2 starting from the initial condition u(0) E IR" at time / = O, we perform K Ktstep prediction of the system state by repeatedly calling the forward evolution net- work Fe with ú((k- + 1) = Fol [k(A) and ú(0) = u()) We then apply the backward evolution network Go to perform K K-step backward prediction: u((k- 1) Go [ū}A)]] and get back to the initial condition (0). This process of forward and backward evolution prediction is referred to as mudti-step reciprocal prediction. The difference between the original value u(O) and the final prediction (()), namely, E [u)] = ||) (()],,, is referred to as the multi-step reciprocal pre- diction error in the Euclidean norm I|I.. In this work, we have the following interesting finding: there is a very strong correlation between the network modeling error {((()) and the multi-step recip- rocal prediction error E (u()). This allows us to use (()()) to approximate the desired network & (u(O)) Note that this multi-step reciprocal prediction
modeling error E [[]( = [Fo(u(()) error can be computed directly with the current state and the wwwarddbackward evolution networks. Its computation does not require the ground-truit truth system state. Therefore, we can use reciprocal prediction error to guide the selection of critical samples from regions with large modeling errors.
Let Jm be the number of samples at m-th iteration of our critical sampling process, current sample set ST = ([u' + u' (A)]: : S S Jm3 is used to train the spatial-temporal evolution network. The corresponding forward-backwant evolution networks are denoted by Fm and G.m. Using the corresponding multi-step reciprocal prediction error distribution (0)),)), we can determine regions sample
with high error values and collect new set of samples S!T which are added to the existing set of samples to update the training set:
sm+
=sp Us ((u((0) + u (A) 1 15 SJS S Jmt1).
(3)
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The above sampling-learninn process is repeated until the overall prediction error drops below the target threshold. According to Table 3 in Supplemental Materials, the complexity for training is increased by 4-5 times, which directly depends the number of iterations needed to reach the threshold for the network modeling error. However, the above critical sample selection process can dramatically reduce the number of needed training samples and related sample collection cost. In the following sections, we will explain this process in more details.
3.2 MULTI- STEP RECIPROCAL PREDICTION ERROR AND CRITICAL SAMPLING
In this section, we show that there is strong correlation between the multi-step reciprocal prediction emrr and the network modeling error of the temporal evolution network Fo"
(1) Multi-step reciprocal prediction. our multi-step reciprocal prediction scheme, we have forward temporal evolution network Fi and backward evolution network 9.' which model the system evolution behaviors in the forward and backward time directions. If the forward and back- ward evolution networks
both well-trained accurately approximating the forward and back- ward evolution operators, for an arbitrarily given system state u(0), the reciprocal prediction error extend this one-step reciprocal pre-
L,s = |] 9 [F [u(O)]] should approach 0. Now, diction to K steps. As illustrated in Figure starting from the initial condition u(0), we repeatedly apply the forward evolution network F8" to perform K-step prediction of the system future states,
ū(kA) = F (k) [u[0]], where F.g (m) represents the k-fold composition of F
F8 Fmm.(k)
F8 o F F0
(4)
k- fold
After K steps of forward evolution prediction, then, starting with ú(KA), we perform K steps of backward evolution prediction using network G ū(kA) g ū(KA)) K 1. 1, 0, where
gm (m-((K-)
9 9
9
(5)
(K -Klk fold
and reach back to time = O. If the forward and backward evolution networks are both accurate, the forward prediction path and the backward prediction path should match with each other. Motivated
by this, we define the multi-step reciprocal prediction error for the forward evolution network Fr as the deviation between the forward and backward prediction paths:
E[u(O)] = Σ ((kA) - ū(k2) 1
(6)
A=
Note that, when computing E[u( we only need the current system state u(0), the forward and backward evolution networks Fr and G Figure 3 shows several examples from the Damped Pendulum and 2D Nonlinear ODE systems listed in Table 2 in Supplemental Materials. The top row shows examples with accurate prediction of their system states. We can see that their forward and backward prediction paths match well and the corresponding multi-step prediction error is very small. For comparison, the bottom shows examples with large prediction errors.
Damped Pendulun
2D Nonlinear
Formad Evolaton Nettooe's Prodiction
Raccboard Evolatin Natookk
Figure 3: Examples of multi-step reciprocal prediction errors on the first two dynamical systems. (2) Predicting the network modeling error. In this work, we find that there is a strong correlation between the network modeling error E [((( and the multi-step reciprocal prediction error (l). Figure 4 shows four examples of E[[(()] (top row) and {()( (middle row) for the Damped
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Pendulum and 2D Nonlinear system with different sizes of training samples. The bottom row shows the values of [0] and E [[] of locations with large errors. We can see that there is a strong correlation between the network modeling error Elu(0)] and the multi-step reciprocal prediction error [0) This correlation allows us to predict El [(] using [[) which can be computed directly from the current system state without the need to know the ground-truth state.
Damped Pendulum
Nonlineur
Nemookk Modeling Eitor Recipooal Prediction Eivor
Corcelaiin Ple
Bongeral Trabosee
Rocpeesl Pooecti tome
Rocgrned Prokzees
Figure 4: Correlation between network modeling error and multi-step reciprocal prediction error on Damped Pendulum and 2D Nonlinear ODE systems.
(3) Critical sampling and adaptive evolution operator learning, Once we are able to predict the network modeling error E()( using the multi-step reciprocal prediction error E[]( we can develop a critical sampling and adaptive evolution learning algorithm. The central idea is to select samples from locations with large values of error []u( using the following iterative peak finding algorithm. Note that u(0) E R", Write u(0) [ui, U2, Un). Let S7 {[u'()) > (A) 1 I < Jm3 } be the current sample set. To determine the locations of new samples, (O) Jm + Sisjj
Jm+1], we find the peak value of multi-step reciprocal prediction error [[](O)] at every sampling point u(0) in the solution space D. In our experiment, we choose the sample point u()) from the augmented sample set Sp defined in (8) of the following section. The corresponding peak location chosen to be tmm+1(00 and the corresponding sample [u'm++ +
u is collected. This process is repeated for Jm+1 Jm times to collect Jm+1 Jmm samples in 2m, which is added to the current sample set:
Sp+1
ST U = (lu' (0) + U (A) 11 Jmtth.
3.3 JOINT SPATIAL DYNAMICS AND TEMPORAL EVOLUTION LEARNING
Joint spatial dynamics and temporal evolution learning aims to couple local dynamics lcarning in the spatial domain and evolution learning in the temporal domain to achieve robust system evolution learning from the small set of selected critical samples.
(1) Sample augmentation based on local spatial dynamics. Let ST = (0) - 15 j J J be the current set of samples collected from the dynamical system. In this work, Jm is small number. For example, in our experiments, Jm is in the range of few hundreds. From our experiments,
find that this learning process is unstable since the number of samples is too small. Our central idea is to design a spatial dynamics network to learn the local spatial dynamics so that we can predict or interpolate more samples at unknown locations from existing samples known locations. Specifically, let (v'(0) E IR < т} be large set of randomly selected points in the state space of IR.. Let (A) & (0)
the system's future state time A when its current state is v'()). Here, {v'(A} are predicted by the spatial dynamics network from the existing sample set Sp denoted by (A) They are added to Sy: as augmentation samples
S ST UVF
VF = [[v'(() ('( 1 S i ('(A) r [S v (0)]},
(8)

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where I (Sp; v' (0)] represent the spatial dynamies network at m-th iteration which predicts the future state of (0) based local spatial change patterns using the existing samples S With this augmented sample set Sy, we can train the temporal evolution network F."".
(2) Learning the local spatial dynamics. The dynamical system may exhibit highly nonlinear and complex behavior in the whole spatial domain, which could be challenging to be accurately modeled and predicted. However, within a small local neighborhood. its behavior will be much simpler and can be effectively learned by our spatial dynamics network Tw. Specifically, given an arbitrary point v(0) = [1,, #2.-.. [n] in R", we find its nearest H points from the existing sample set ST, and the corresponding samples are denoted by Svro) = [[2" (0) + z s h S H), which are the input to the spatial dynamics network. We use a pth-order n-variate polynomial P(u)- P(c C2. Cu,( ... , 2n to locally approximate the local spatial dynamics. The coefficients of the polynomial are [ct, C2 ,p,,, which are predicted by the spatial dynamics network Tw. w. For example, if 1, this becomes linear approximation with P = n+1 coefficients. p = 2, the number of coefficients, or the size of the network output becomes P = ( ( (n +). To summarize, the task of the spatial dynamics network T.. is to predict the coefficients of the polynomial P(u) from the set of M neighboring samples (v so that P P(u)lu=v()) = v(A), where
(A) is the future state of the system at time A when its current state is v(0), or ( = (v(()))
When training the spatial dynamics network I on can choose the sample from the existing sample set ST as the input (0) and the corresponding output as v(A). The L2 loss between the predicted state at time A and its true value for v(),, namely,
LSDN Σ
((A) P SS F v(o]
v(OJESF
(3) Joint learning of spatial dynamics and temporal evolution networks. The temporal evolution network Fo and the spatial dynamics network TT aim to characterize the system behavior from two different perspectives, the temporal and spatial domains. In this work, we couple these two networks so that they can learn more effectively. Specifically, we use the spatial dynamics network n to generate a large set of samples Vr, which is added to the existing samples Sy , as explained in (8) This augmented sample set Sp: is used to train the temporal evolution network F." Note that both networks are predicting the system future state from spatial and temporal domains. Therefore, we can introduce consistency constraint between them. Specifically, let Q {qr E RR" 1 < 1 < L} be a set of randomly generated points in R". We use both networks to predict the future state at time A for each q at the initial state. The following consistency loss is used to train both networks:
L
F [q]]
r
; q:)
(10)
3.4 THEORETICAL UNDERSTANDING
This section provides some mathematical analysis results to understand and characterize the perfor- mance of the proposed critical sampling and adaptive evolution learning method.
Assumption 3.1. Assume that the underlying dynamical system म utt) = H(u(t)) is autonomous with H. being Lipschitz continuous (the Lipschitz constant denoted by C+x) on a set D C R.".
which is a bounded region where we are interested in the solution behavior.
Assumption 3.2. Suppose that the generalization error of the trained neural network is bounded: 4 (
||Fe
Ef < +00, Go WAllE>(D) =: Eg +00.
(11)
We derive a generic bound for the prediction error of our network model. Let ((k) be the forward predicted solution by the trained primal network model at time (K) to AA starting from to and (4) be the backward predicted solution by the trained dual network model at time (()) starting from (KK), where 0- < k < K. Denote the corresponding forward prediction error as ((t) :=
( ((((())]] and the backward prediction error as E(k) == ( u(:(k))],, k = 0, K. Theorem 3.3. Under Assumptions 3.7 and we have the following estimates.
(1) ((k), u(((+)) E Do 0 < S K with DA = (uE D: (u) E D A7 E A]} then
c Cukks -1
{(k) < CNKA, +
cCHd- Cg-
(12)
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(2) if ()) E Da for 1 ks K with DA == {u D: ti (u) E D Vt E [0, then (C((K=N)
g(k) v ex
ccHa-1
(13)
Please see Section C for the detailed proof of this theorem and further discussions in Theorem C.4 The estimates (12 imply that in the worst cases the prediction errors may grow exponentially with the number of steps k. The analysis suggests that the reciprocal prediction error correlated with the network modeling error, which provides theoretical support for our finding. The critical sampling can help to effectively reduce the reciprocal prediction error and suppress the undesirable error growth, thereby enhancing the accuracy and stability of our model.
4 EXPERIMENTAL RESULTS
4.1 EXPERIMENTAL SETTINGS
We follow the evaluation procedure used in existing research, for example those reviewed in Section 2. to evaluate the performance of our proposed method on specific examples of dynamical systems. We consider four representative systems with ODEs and PDEs as their governing equations, as summarized in Table 2 in Supplemental Materials. They include (1) the Damped Pendulum ODE equations in TR?2, (2) a nonlinear ODE equation in TR2, (3) the Lorenz system (ODE) in TR3, and (4) the Viscous Burgers' equation (PDE). Note that, for the final PDE system, we approximate it in generalized Fourier space to reduce the problem to finite dimensions as in Wu & Xiu (2020). We the projection operator P, V Vn, where V, span fsin (jz) J & n2 with Certainly, our proposed method can be also applied to many other dynamical systems, we simply use these four example systems to demonstrate the performance of our new method. In Section AA. we provide detailed descriptions on how to obtain the training samples for the dynamical systems. An overview and pseudo-code of our method can be found in Section A.4
9.
4.2 PERFORMANCE RESULTS
We choose the evolution learning method developed in Qin et al. 2019 Wu & Xiu (2020) as our baseline. This method has achieved impressive performance in learning the evolution behaviors of autonomous systems and attracted much attention from the research community. On top of this method, we implement our proposed method of critical sampling and adaptive evolution learning- We demonstrate that, to achieve the same modeling error, our method needs much fewer samples.
Table I compares the numbers of samples needed for learning the system evolution by the baseline method and our critical sampling and adaptive learning method. For example, for the Lorenz system, needs 1000000 samples to achieve the modeling error of 0.197. Using our proposed critical sampling method, the number of samples can be reduced 1765, while achieving even smaller modeling error 0. 194 The number of samples has been reduced by 567 times.
Table 1: Samples for learning the system evolution using the baseline method and our method, The prediction errors are evaluated on 50 arbitrarily chosen solution trajectories in the computational domain. All testing states are not included in the training set. Average error and standard deviation reported for each dynamical system. See Section A.3 in Supplemental Materials for more details.
Baseline
Our Work
Dynamical System
Ratio
Samples
Prediction Error
Samples
Prediction Error
Damped Pendulum
14400
0.02630 1 0.01200
417
0.02411
0.00991
34.53
2D Nonlinear
14400
0.00037
0.00021
925
0.00035
0.00015
15.57
Lorenz System
1000000
0.19685 + 0.07768
1765
0.19357
0.05695
566.57
Viscous Burgers' Eq.
500000
0.01679 + 0.00878
19683
0.01652 + 0.00818
25.40
Figure 5 shows the performance comparison results for the four dynamical systems listed in Table
2 In each sub-figure, the horizontal dashed line shows the average network modeling error achieved by the baseline method for the number of samples shows in the legend. This number is empirically chosen since it is needed for the network to achieve reasonably accurate and robust learning per- formance. We can see that as more and more samples are selected by our critical sampling method, the network modeling error quickly drops below the average modeling error of the baseline method.
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Damped Pendulum
2D Nonlinear
Lorenz System
Viscous Burgers' Equation
Rasidl Sarpl
Nanved Madeloo Flove
Rand Soeple
Natvec Medidia
Mudeiing
Nurther Sanglee
Narther of Samptic
Narbber Sumppie
Narber of Surppic
Figure 5:
The critical sampling and adaptive learning results four dynamical systems.
Figure 6 shows that the network modeling error {(] of the Damped Pendulum system, is being quickly reduced with more and more critical samples are collected. In Figure 7 the top row shows the phase portraits of the solutions for two systems, Damped Pendulum (left) and 2D Nonlinear system (right), obtained by our method with comparison against the ground-trute solutions. The bottom row shows an example solution for the Viscous Burgers' PDE system. The first one the ground-truth solution. The second and third figures are the solution and its difference from the ground-truth solution for the baseline method. The last two figures are the solution and difference obtained by our method. We can that using fewer samples, evolution and predict its future states at the same level of accuracy. In Section D we provide more experimental results to further understand the performance of proposed method.
method is able to learn the system
225 Samples
297 Samples
361 Samples
417 Samples
Figure 6: The reduction of network modéling error with more and more critical samples are collected for the Damped Pendulum system.
Ground Trufi
Our Method with 417 Sumples
Goannd Trath
Our Methoa with 925 Sumples
Feroue Preietin
Basclin: Mehhod 500000 Sumples
Methed Samples
Figure 7: Solutions obtained by critical sampling and adaptive evolution learning method.
CONCLUSION AND FURTHER DISCUSSION
In this work, we have studied the critical sampling for adaptive evolution operator learning problem. We have made an interesting finding that the network modeling error is correlated with the multi-step reciprocal prediction error. With this, we able perform dynamic selection of critical samples method
from regions with high network modeling errors and develop an adaptive for dynamical systems based on the spatial-temporal evolution network. Extensive experimental results demonstrate that our method is able to dramatically reduce the number of samples needed for effective learning and accurate prediction the evolution behaviors.
In the future, we hope to apply our approach to large-scale dynamical systems, by combining some reduced-order modeling or lifting techniques (cf. Qian et al. 22200) or incorporating certain sparsity (cf. SShaaeffer et al. (2018)) Another important question that has not been fully addressed in this paper is how to control the system state towards those samples selected by our critical sampling method. During simulations, this system state control is often available. However, for some complex systems, the exact change of the system state may not be trivial. this case, we shall investigate how the system state control impacts the critical sampling and system modeling performance.

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REPRODUCIBILITY STATEMENT
We provide detailed data generation process and experiment settings in Section A of Supplemental Materials. An overview and pseudo-code of our method can be found in Section A.4 of Supplemen- tal Materials.
REFERENCES
Josh Bongard and Hod Lipson. Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 104((4):9943 99999999999999999 2007.
Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932-399) 2016.
Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, Eurika Kaiser, and Nathan Kutz. Chaos as an intermittently forced linear system. Nature Communications, 8(1):1-9, 2017.
Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary dif- ferential equations. In Advances in Neural Information Processing Systems, 2018.
Zhen Chen and Dongbin Xiu. On generalized residual network for deep learning of unknown dy- namical systems. Journal of Computational Physics, 438: 10362, 2021.
Zhen Chen, Victor Churchill, Kailiang Wu, and Dongbin Xiu. Deep neural network modeling of unknown partial differential equations in nodal space. Journal of Computational Physics, 449: 110782, 2022.
Emily Fox, Erik Sudderth, Michael Jordan, and Alan Willsky. Nonparametric bayesian learning of switching linear dynamical systems. In Advances in Neural Information Processing Systems, volume 21, 2008.
Andrew M. Fraser. Hidden Markov Models and Dynamical Systems. SIAM, 2008
Xiaohan Fu, Lo-Bin Chang, and Dongbin Xiu. Learning reduced systems via deep neural networks with memory. Journal of Machine Learning for Modeling and Computing, 1(2), 2020.
Yuanjun Gao, Evan W. Archer, Liam Paninski, and John P. Cunningham. Linear dynamical neural population models through nonlinear embeddings. In Advances in Neural Information Processing Systems, volume 29, 2016.
Min Han, Zhiwei Shi, and Wei Wang. Modeling dynamic system by recurrent neural network with state variables. In International Symposium on Neural Networks, pp. 200-205. Springer, 2004.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp- 770-778, 2016.
Yu Huang, Yufei Tang, Xingquan Zhu, Hanqi Zhuang, and Laurent Cherubin. Physics-coupled spatio-temporal active learning for dynamical systems. IEEE Access, 2022.
Masanobu Inubushi and Kazuyuki Yoshimura. Reservoir computing beyond nemory-nonlineee trade-off. Scientific Reports, 7(1):1-10, 2017.
Matthew J. Johnson, David K. Duvenaud, Alex Wiltschko, Ryan Adams, and Sandeep R. Datta. Composing graphical models with neural networks for structured representations and fast infer- ence. In Advances in Neural Information Processing Systems, volume 29, 2016.
Rachael T. Keller and Qiang Du. Discovery of dynamics using linear multistep methods. SIAM Journal on Numerical Analysis, 59(1):429 455, 2021.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In International Conference Learning Representations, 2021a.
10

---

Under review as a conference paper at ICLR 2023
Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli,, and Anima Anandkumar. Physics-infe informed neural operator for learning partial differential equations. arXiv preprint arXiv:2111.03399 2021b.
Scott Linderman, Matthew Johnson, Andrew Miller, Ryan Adams, David Blei, and Liam Paninski. Bayesian learning and inference in recurrent switching linear dynamical systems. In Artificial Intelligence and Statistics, pp. 914-922. PMLR, 2017.
Artémis Llamosi, Adel Mezine, Florence d' Alché Buc, Véronique Letort, and Michele Sebag. Ex- perimental design dynamical system identification: a bandit-based active learning approach. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp- Springer, 2014.
Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong. Pde-net: Learning pdes from data. In International Conference on Machine Learning, pp. 3208-3216. PMIR, 2018.
Zichao Long, Yiping Lu, and Bin Dong. Pde-net 2.0: Learning pdes from data with a numeric- symbolic hybrid deep network. Journal of Computational Physies, 399:108925, 2019.
Niall M. Mangan, J. Nathan Kutz, Steven L. Brunton, and Joshua L. Proctor. Model selection for dynamical systems via sparse regression and information criteria. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2204):201 70009, 2017.
Horia Mania, Michael I. Jordan, and Benjamin Recht. Active learning for nonlinear system identi- fication with guarantees. Journal of Machine Learning Research, 23:32-1, 2022.
Kevin McGoff, Sayan Mukherjee, and Natesh Pillai. Statistical inference for dynamical systems: A review. Statistics Surveys, 9:209-252, 2015.
Lev Semenovich Pontryagin. The mathematical theory of optimal processes. Interscience, 1962.
Elizabeth Qian, Boris Kramer, Benjamin Peherstorfer, and Karen Willcox. Lift & learn: Physics- informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, 406: 32401,, 2020.
Tong Qin, Kailiang Wu, and Dongbin Xiu. Data driven governing equations approximation using deep neural networks. Journal of Computational Physics, 395:620-665 635, 2019.
Tong Qin, Zhen Chen, John D. Jakeman, and Dongbin Xiu. Data-driven learning of nonautonomous systems. SIAM Journal on Scientific Computing, 4313):11077A1122 2021a.
Tong Qin, Zhen Chen, John D. Jakeman, and Dongbin Xiu. Deep learning of parameterized equa- tions with applications to uncertainty quantification. International Journal for Uncertainty Quan- rification, 11(2), 2021b.
Lawrence R. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):557-26,, 1989.
Maziar Raissi. Deep hidden physics models: Deep learning of nonlinear partial differential equa- tions. Journal of Machine Learning Research, 19(1):932-95 2552 2018.
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Machine learning of linear differential equations using gaussian processes. Journal of Computational Physics, 348:683-693, 2017.
Maziar Raissi, Paris Perdikaris, and George Karniadakis. Physics-informee neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019.
Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Data-driven discovery of partial differential equations. Science Advances. 3(4):e1602614 2017.
Hayden Schaeffer and Scott G. McCalla. Sparse model selection via integral terms, Physical Review E, 96(2))02330,, 2017.
11

---

Under review as a conference paper at ICLR 2023
Hayden Schaeffer. Giang Tran, and Rachel Ward. Extracting sparse high-dimensional dynamics from limited data. SIAM Journal on Applied Mathematics, 78(6):3279- 3295, 2018.
Michael Schmidt and Hod Lipson. Distilling free-form natural laws from experimental data. Science, 324(5923):81-83 2009.
A. Stuart and R. Humphries. Dynamical Systems and Numerical Analysis. Cambridge University Press, 1998.
Wei-Hung Su, Ching-Shan Chou, and Dongbin Xiu. Deep learning of biological models from data: Applications to ode models. Bulletin of Mathematical Biology, 83(3):1-19, 2021.
Pantelis R. Vlachas, Wonmin Byeon, Zhong Y. Wan, Themistoklis P. Sapsis, and Petros Koumout- sakos. Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, (742213):201708444 2018.
Kailiang Wu and Dongbin Xiu. Numerical aspects for approximating governing equations using data. Journal of Computational Physics, 384:200-221, 2019.
Kailiang Wu and Dongbin Xiu. Data-driven deep learning of partial differential equations in modal space. Journal of Computational Physics, 108:10
Kailiang Wu, Tong Qin, and Dongbin Xiu. Structure-pressrriing method for reconstructing unknown
hamiltonian systems from trajectory data. SIAM Journal on Scientific Computing, 42360 A3729, 2020.
Hao Xu and Dongxiao Zhang. Robust discovery of partial differential equations in complex situa- tions. Physical Review Research, 3(3):033270, 2021.
Hao Xu, Haibin Chang, and Dongxiao Zhang. Diga-pde: Discovery of pdes with incomplete can- didate library via combination of deep learning and genetic algorithm. Journal of Computational Physics, 418: 109584, 2020.
Hao Xu, Dongxiao Zhang, and Junsheng Zeng. Deep-learning of parametric partial differential equations from sparse and noisy data. Physics of Fluids, 33(3):037132, 2021.
Chi Zhang and Abdollah Shafieezadeh. Simulation-free reliability analysis with active learning and physics-informed neural network. Reliability Engineering & System Safety, 226:108716, 2022.
Juntang Zhuang, Nicha Dvonekk, James Duncan, al. Mali: A memory efficient and reverse accurate integrator for neural odes. In International Conference on Learning Representationss 2020a.
Juntang Zhuang, Nicha Dvorekk, Xiaoxiao Li, Sekhar Tatikonda, Xenophon Papademetris, and James Duncan. Adaptive checkpoint adjoint method for gradient estimation in neural ode. In International Conference on Machine Learning, pp. (669--16699 PMLR, 2020b.
Christoph Zimmer, Mona Meister, and Duy Nguyen-Tuood Safe active learning for time-series modeling with gaussian processes. In Advances in Neural Information Processing Systems, vol- ume 31, 2018
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SUPPLEMENTAL MATERIALS
In Supplemental Materials, we provide more algorithm implementation details, further discussion on related work, further analytical understanding, and additional experimental results for in-depth understanding of our proposed method.
A EXPERIMENTAL SETTINGS AND IMPLEMENTATION DETAILS
In this section, we provide more details system configurations and algorithm implementation.
A.1 DYNAMICAL SYSTEMS STUDIED IN THIS WORK
In this paper, we consider four representative systems with ODEs and PDEs as their governing equations. They include (1) the Damped Pendulum ODE in R2, (2) nonlinear ODE system in IR, (3) the Lorenz system (ODE) in R and (4) the Viscous Burgers' equation (PDE). Their governing equations are shown in Table
Table 2: Overview of the governing equation systems demonstrate in this work.
SYSTEM
GOVERNING EQUATIONS
d R u2
DAMPED PENDULUM EQUATION
de u2
0.2u2 8.91 sin u1.
d
(ul + u2 1) U1
A 2D NONLINEAR EQUATION
dt
u2
de
d
U2
u2 111 2 + U2 2 1).
d U1 10 (u2 11
LORENZ SYSTEM
de U2 u1 (28 us) u2
Ug U1t/2 (8/3)ts u33
VISCOUS BURGERS' EQUATION
Mr +
3/
0.1umzi (-,
u(-TT!) "u(T, 1) = 0, LERt.
A.2 SYSTEM CONFIGURATIONS
For the ODE examples, we follow the procedure Qin et al. (2019) to generate the training data pairs (0), u² (A]]} follows. First, we generate J system state vectors {u' (0)) based on a uniform distribution over a computational domain D. Here, D is the region where we are interested in the solution space. It is typically chosen to be hypereube prior to the computation, which will be explained in the following. Then, for each j, starting from u² (0), we solve the true ODEs for time lag of A using highly accurate ODE solver to generate u' (A). Notice that, once the data is generated, we assume that the true equations are unknown, and the sampled data pairs the only known information during the learning process.
j= at time 0
For the first example dynamical system listed in Table its computational domain is = [-Tr, π x
[-2x, 2T). We choose A = 0.1. For the second system, the computational domain is D = 21 The time lag A is set 0.1. 1. For the third system, the computational domain is D [ 25)2 x [0, 50). The time lag is set as 0.01.
For the Viscous Burgers' PDE system, because the evolution operator is defined between infinite dimensional spaces, and we approximate a modal space, namely, a generalized Fourier space, in order to reduce the problem to finite dimensions as in Wu & Xiul (2020). We follow the same
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procedure specified in Wu & Xiu (0200) to choose basis of the finite dimensional space V, to represent the solutions, then apply the projection operator to project the snapshot data to Vn to obtain the training data in the generalized Fourier space. The choice of basis functions is fairly flexible, any basis suitable for spatial approximation of the solution data can be used. Once the basis functions are selected, projection operator Pa V + Vn is applied obtain the solution in the finite dimensional form.
The approximation space chosen to be relatively larger as V, = span {sin (jx) : 1 1 n) with n = 9. The time lag A is taken as 0.05. The domain D in the modal space is set as [-1.5, 1.5] x [-0.5, 0.5] 0.2, 0.2]² x 0.1) x 0.00, 0.05]² x [-0.02, 0.02], from which we sample the training samples.
Our task is to demonstrate how our proposed method is able to significantly reduce the number of samples needed for evolution learning. Specifically, for the baseline method, we use random sam- pling, randomly selecting locations in the solution space to collect samples for evolution learning. For example, for the first dynamical system. Damped Pendulum system (ODE) in a 2-D space, the baseline method use 14400 samples to achieve an average network modeling error of 0.026. We then use our method to adaptively discover critical samples and refine the evolution network to reach the same smaller network modeling error. We demonstrate that, to achieve the same modeling error, our proposed method needs much fewer samples.
A.3 IMPLEMENTATION DETAILS
In all examples, we use the recursive ResNet (RS- Res-RRsNett architecture in He et al. 2019), which a block variant of the ResNet and has been proven in Qin et al. (2019); Wu Xiu 2020 to be highly suitable for learning flow maps and evolution operators.
(2016); Qin et al.
For all the 4 systems, the batch size is set as 10. In the two 2-dimensional ODE systems, we use the one-block ResNet method with each block containing 3 hidden layers of equal width of 20 neurons, while in the 3-dimensional ODE system, we use the one-block ResNet method with each block containing 3 hidden layers of equal width of 30 neurons. For the final PDE system, we the four-block ResNet method with each block containing 3 hidden layers of equal width of 20 neurons. Adam optimizer with betas equal (0.9, 0.99) is used for training. In the two 2-dimensional ODE systems, all the networks are trained with 150 epochs. In the Lorenz system and Viscous Burgers' equation, all the networks are trained with 60 epochs. The initial learning rate is set as 10 and will decay gradually to 10-0 during the training process. All networks are trained using Py Torchh on one RTX 3060 GPU.
In the four example systems, we evaluate the performance of our models on time duration t E [0, 20), I E [o. 10), / E [0, 5). I E [0, 2). respectively. For the first two ODE systems, the network modeling emorr is evaluated by average MSE enror at each time step on 50 different arbitrarily chosen solution trajectories. For the Lorenz system, we evaluate the network by average MSE error each time step under 50 different initial conditions. For the final PDE system, the network modeling error evaluated by the average L2 norm error on 100 points at time = 2 under 50 different initial conditions.
A.4 PSECDO-CODE AND OVERVIEW OF OUR PROPOSED METHOD
Our proposed method of critical sampling and adaptive evolution operator learning algorithm is summarized in Algorithm [] Our proposed method has the following steps: First, we generate Jmm data pairs based on a uniform distribution over a computational domain D using highly accurate ODE/PDE solver. Then, we train the spatial dynamics network I using the generated data pairs and use the network to generate a large set of additional samples. After that, we train forward evolution network Fil using the data pairs, and train backward evolution network G using the reversed data pairs. Multi-step reciprocal prediction error is evaluated on different locations in computational domain D with both networks. Finally, we collect the samples from the location with peak reciprocal prediction error. Those samples should be added to the initial set and the whole process is repeated until the network modeling error [n) is smaller than threshold.
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Algorithm 1: Critical Sampling and Adaptive Evolution Operator Learning Algorithm
Input: Number of samples in the initial set Jmi Number of samples in the updated set Jm+11 Batch size; Learning rate; Adam hyperparameters; Initial parameters of networks. Output: Optimized forward evolution network Fo-
Generate Jm data pairs based on a uniform distribution over computational domain D, initialize the sample set S {[u (0) (')) 1s S Jm
repeat
// Sample augmentat ion based local spatial dynamics. I using Sy:
Train spatial dynamics network
Use Tw generate large set of samples Vr, add to the existing sample set:
Sp - Sp UVF:
// Multi-step reciprocal prediction.
Reverse the data pairs in ST to get Sy:
Train forward evolution network F using Sp. backward evolution network GT using St Perform K-step forward prediction using F4 get úkkA);
Perform K Ksttep backward prediction using 97 to get u(ká):
Calculate multi-step reciprocal prediction error E[u(()] Σk (k(A)) a(k>)))
// Critical sampl 1 ing. Initialize critical sample set 12n to empty set; for i in (1.2.... Jm+1 Jm3 do
10
11
Choose location with peak reciprocal prediction error B[u(0)] to be (0)=
Collect corresponding sample (1==+((0) (0) u (A)), add to Ωm
14
end
15
Add the critical sample set to the current sample set (without augmented samples Vp)):
SF Sm++1 = Sp Usm
ST sm+1 Jm Jm+
16
until network modeling error Elu(O)) smaller than threshold:
17
B FURTHER DISCUSSION ON RELATED WORK AND OUR UNIQUE CONTRIBUTIONS
For the data-driven discovery of ODEs and PDEs, there are two major popular approaches. The first approach typically tries to identify all the potential terms in the unknown governing equations
from a priori dictionary, which includes all possible terms that may appear in the equations (Brunton
et al. 2016 Schaeffer & McCalla 2017 Rudy al. 2017 Raissi 2018 Long al.. 2018 Wu &
Xiul 2019 Wu et al.) [020 Xu & Zhang. 2021). The second approach for data-driven learning of
unknown dynamical systems is to approximate the evolution operator of the underlying equations.
instead of identifying the terms in the equations (Qin et al. 2019 Wu & Xiu 2020 Qin et al. 202 la
Chen et
2022 Li Li al. 2021b).
In this work, we choose the second approach of learning the evolution operator. Compared to the first approach which tries recover the mathematical expression of the unknown governing equations, the second approach of learning evolution operators often has the following distinctive features:
(1) does not require a prior knowledge about the form of the unknown governing equations. (2) Unlike the first approach, the evolution operator approach, which is based on the integral from of the underlying dynamical system, does not require numerical approximation of the temporal derivatives and allows large time steps during the learning and prediction process. (3) Although the first approach may successfully recover the expressions of the governing equations, during the prediction stage, it still needs to construct suitable numerical schemes to further solve the learned equations. On the contrary, the second approach learns the evolution operator which can be directly used perform long-term prediction of the system behavior in the future.
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Learning deep neural network model for the evolution operator often requires very large number of training (often in the ranges of tens or hundreds of thousands), especially for nigh-dimensional dynamical systems. Existing methods in the literature often assume that samples always avail- able. In practice, this is not the case. In field conditions of dynamical systems, sample collection is often costly or very limited due to resource constraints experimental accessibility. To our best knowledge, this work is one of the first efforts to address this challenge.
For bandwidth-limited signals, the Nyquist sampling theorem can be used to determine the mini- mum number of samples needed to perfectly reconstruct the original signals. For sparse signals, compressive sensing theorems have been developed in the past years to determine the number of ob- servations needed to reconstruct the original signals. However, for dynamical systems with highly nonlinear and dynamic behaviors, it is very challenging to develop corresponding analytical results for minimum sampling. This task becomes even more challenging for systems whose governing laws or equations are unknown to us.
In this work, instead of pursuing analytical approach to answer the above critical sampling ques- tion, we propose to explore design-based approach which aims to address the following issues: (1) how to design a deep neural network for dynamical systems which robustly learn its evolution behavior with a smaller set of samples and effectively improve its prediction accuracy with more and more critical samples being added? (2) How to design algorithm to dynamically determine the most important samples or critical samples for network model learning? (3) How can we couple the deep neural network model learning process with the critical sampling algorithm to establish an adaptive joint campling-learninn method for dynamical systems to drastically reduce the number of needed samples for effective learning of the system evolution behavior?
Although our paper and Qin et al. (2019); : Wu & Xiu (2020) study the same topic "learning the evolution behavior of unknown dynamical systems", our paper focuses on investigating a different yet very important problem: how to learn the evolution behavior of unknown dynamic systems with very few samples and how to dynamically select these samples. This paper has significant impacts in practice since collecting samples from real-world dynamical systems can be very costly or very limited due to resource/labor constraints or experimental accessibility.
The success of our approach also contributes significantly to deep learning and signal estimation research. It investigates an important problem in learning and prediction with deep neural networks: how to churacterize and estimale the prediction errors during the inference slage? This is a very challenging problem since the ground -ruth values for unknown dynamical systems are not available.
This problem not studied in Qin et al. (019): Wu & Xiu (2020) which present a ResNet ap- proach to model the evolution behavior of the dynamical systems. Those methods typically assumed availability of sufficient data, and often used tens of thousands even millions of samples for train- ing.
When addressing these important problems, paper has the following unique novelties and con- tributions: First, we have introduced the concept of multi-step reciprocal prediction error based on a prediction loop of forward and backward deep neural works. When modeling the evolution behavior of unknown dynamical systems, we discovered that the prediction error of the deep neural networks highly correlated with the multi-step reciprocal prediction error. Second, most interestingly, un- like existing methods in prediction error estimation which often replies on assumptions about the underlying data statistics or distributions. Our method computes the prediction error based on the single input data itself. Third, we have introduced the approach of work learning approach which couples temporal evolution learning with spatial dynamics learning to significantly improve the learning performance with very few samples.
evolution net-
The adjoint state method (Pontryagin) 1962) is numerical method for efficiently computing the gradients of functions for numerical op- timization problems. Recently, it has been successfully used for highly efficient learning the ODE for dynamical systems (e.g.. (Chen al. 2018, Zhuang et al.) 2020b ab). It has also been further extended using the adaptive check point method for more accurate estimation of the gradients for neural ODE (Chen et al.) 2018). To address the memory cost and the inaccuracy in reverse-ting time trajectory, a memory-efficient ALF Integrator (MALI) has been developed (Zhuang et al. 2020a). Our multi-step forward and backward processes and the usage of mismatch errors adjust the esti- mation of our proposed method look similar to those in the recent adjoint state methods (Chen et al.
Relations to and Differences from Adjoint State Method.
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2018 Zhuang al. 2020a b). However, method is uniquely different in the following aspects. (1) In the adjoint state method, the backward propagation is performed for computing the gradients of loss function. However, in method, the backward network is for learning the inverse opera- tor of the forward evolution operator (namely, the backward evolution operator, please see Lemma C.T). (2) The adjoint state method aims compute the gradients of the loss functions more effi- ciently and more accurately. However, our method aims to discover the critical samples for network learning. The purpose of our multi-step reciprocal error using the forward and backward networks is to estimate the network prediction error. We have made very interesting finding: this multi-step reciprocal error is highly correlated with network prediction error. Note that this prediction error cannot be obtained since the ground-truth is not available. This finding is very important in our network learning and prediction. also allows us to perform critical sample discovery for unknown dynamical systems and learn the system behavior with very few samples. Once the critical samples are discovered, the network is trained as usual. No adjoint state methods are used here to compute the gradients during the training process.
C FURTHER ANALYTICAL UNDERSTANDING OF OUR PROPOSED METHOD
In this section, we provide further analytical understanding of our proposed method of critical sam- pling and adaptive evolution learning, especially on the error bound of evolution operator learning. Let us consider the following autonomous ODE system as example:
du(() H(u(t)), t E
(14)
dt
where u(t) R the state variables. Let $ R" + R" be the evolution operator, which maps the system state from time = 0 to its next state at time A: u(t + A) = (u()). It should be noted that, for autonomous systems, this evolution operator 4 remains invariant for different time instance f. It only depends on the time difference A.
Lemma C.1. The backward evolution operator V of system (4) is actually the forward evolution operator of the following dynamical system
dt i()) = -H(((())-
(15)
Proof.
Define u(l) = u(T -t) for an arbitrarily fixed T >>> 0. can be seen that
d
du
di a(t)
(u(T - ()) =
= --((u(T-))) - -HH((()).
di
di T-t
This means i(t) satisfies the ODEs 15) The forward evolution operator of system 15 which maps a(t) ū(t + A), equivalent the mapping from u(T -t) 1) to u(T - A), which exactly
coincides with the backward evolution operator V of system [4 The proof is completed.
In the following analysis, will always use Assumption 3.1 with H. being Lipschitz continuous (the Lipschitz constant is denoted by Cx) on a set C R Here D is bounded region where we interested in the solution behavior.
Lemma C.2. Suppose Assumptions
3.1 holds. Define
DA {u LuEE:: D: (u) D Vt [0,
The forward evolution operator & of system 14 is Lipschitz continuous Ds i.e. for any
U1 U2 E DA
(u)) ( - CnA uzll-
(16)
Proof. This follows from a classical result in the dynamical system; see (Stuart Humphries 1998).
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Lemma C.3. Suppose Assumptions 3.7 holds. Define
DA {u D: V2(u) D Vt E [0,
The backward evolution operator V a system
14) is Lipschitz continuous on Da iee, for any
u1, U2 E Da.
| (n
A (12)
cx In
uzll-
(17)
Proof. According to Lemma C.1 VA is the forward evolution operator of system 15
Following
the idea of Lemma C.2 for A one can complete the proof.
We now derive simple generic bound for the prediction error of our network model. Suppose the
generalization eπror of the trained neural network is bounded: Let (()) be the forward predicted solution by the trained primal network model at time (()) ;= to + Kå starting from to. and ( be the backward predicted solution by the trained dual network model at time t(() starting from {(K), where 0 v k v K. Denote the corresponding forward prediction error as ((k) == [] - (
and the backward prediction error as g(k) == [| (l) k 0, l..., K. We then have the following estimates.
Theorem C.4. Under Assumptions 3.7 and 3.2
we have:
1. i u(((()) E Da k then
&Cuhs -1
E(R) S (00) Cuks +
cCuA - 68.
(18)
2. ff ( E for S k 1, then
-1
g(R) < g(K) Cu(K))(
-Cnn-1 -1 Eg.
(19)
3. Furthermore, if we take ú0)) u(to) and pass ū(K) as the input of the trained dual network with (K) a(K), then
(Crhss -1
800) S minn
l x(K-N) -1 (=(D)) +
pons_ -1 Eg,
(20)
e cCcKA-11 -1)
(Cnik--bj--)
EgE Cx(K-()))
eOns -Cn-11
Eg
and in particular when k =
u(to)ll2
min { |105 F(( 1 1lo
(CCcks-) (1
(oGnnks+e))
(21)
where denotes the identity map.
Proof. Recall that (*) (u 1) and (()( = 1 for the Euclidean norm, the assumption (1). and the Lipschitz continuity (16) of & in Lemma we can derive that
a (u()(t))))), Using the triangle inequality
{(k) (1) - 1000-u(((()) 1(((1))
Fo 1)) 4 (u((((t1)))
< Fo(ū(h-1)) T ( |)
+ (n(k-11) - a a ((((((e11))) $ |F - - (I> + CNs (a=-1) u(t(a-1))/
Ond +
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Under review as a conference paper at ICLR 2023
Repeatedly utilizing such estimate leads to
{(h) eCn te s ( + e eg tef
g 20n ej e en +
k-1 k ((0) t Σ sons
which completes the proof of 18
Similarly, for the backward prediction procedure, we recall that i (k) -G( and u(t (16) v (a(u(t (((f(+++)) and then use Lemma C.3 to deduce that
(M) - 1((( g(k)
(o(ū((+1)) - v (a(
Go (((k)t)) - V + ( - v V (
< 1190 Vall + e ens (((tt)) u(l(kt2))))
Exdg((++1) + cg.
Repeatedly utilizing such estimate leads to
g(k) S e Cn + cg
I e 20 g(k+2) + e CN Cx tgg
s u0n g(kt3) 20n5 on e en cgg
K-k-1 S (K ())x) +
Σ Jnns
which completes the proof of 19 of the trained dual network with (K) (K), then (()) and {Kk {(K), Combining (18) and (19 gives
Furthermore, we take ú ()) u(to) and pass (k) as the input
{K)) S ((K) (x()))((K- + (cC((K-MAA-1) cg
CCnKA-1 - sreCn(K-))s +
(x((K
CCAA -1
Eg.
(22)
On the other hand, we observe that
(k)
i n n((()))
(k) ń (1(33) 1//011 + +1/1(()) u(((())
ofk (K--)) x)), F/ (k) -i( + g(k)
| (K-- xk) =((-() -1 L>(D)
+ t(K))
@ocks -
- o (K-R x) F((K -1
L>(D)
+
xCHA_
Ex,
(23)
where we have used the triangular inequity for the Euclidean norm and the estimate 23 with (22) gives 20) and completes the proof.
[8].
Combining
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Under review as a conference paper at ICLR 2023
Table 3: Training time and inference time of our method and baseline method in all the experiments.
SYSTEM
METHOD
SAMPLES
TRAINInG TIME (s)
INFERENCE TIME (S)
OURS
417
2554.8
0.314
DAMPED PENDULUM
BASELINE
14400
749.7
0.296
OURS
925
1868.4 4
0. 0.121
2D NONLINEAR
BASELINE
14400
444.1
0.1
OURS
1765
81473.0
0,403
LORENZ SYSTEM
BASELINE
1000000
14936.1
0.384
OURS
19683
68263.6
0.483
VISCOUS BURGERS'
BASELINE
500000
12137.5
0.488
D ADDITIONAL EXPERIMENTAL RESULTS AND PERFORMANCE
COMPARISONS
D.1 EXPERIMENTS ON COMPUTATIONAL COMPLEXITY
we present the training time and inference time of our method and baseline method spectively in all 4 numerical experiments. It shows that our method requires more time on training. Although the proposed method has higher complexity, our method can dramatically reduce the num- ber of needed samples, therefore, reducing the sample collection cost. This is highly important in practice, especially with real-world dynamical systems.
In Table
D.2 ADDITIONAL EXPERIMENTAL RESULTS
We conduct experiments on the first Damped Pendulum system. Using our critical sampling and adaptive evolution operator learning method, the number of training samples needed for evolu- tion behavior learning reduced significantly from 14,400 417. Figure R compares the phase plots for the Damped Pendulum Equation with an arbitrarily chosen initial state u(t 0) (-1.193, 3.876) predicted by the baseline method with 14,400 samples and our method with 417 samples. In each sub-figure, the orange line represents the ground-truth solution, and the blue dotted line shows the network prediction results. In Figure 9 the left column shows the trajectories for (M1, u2) 2 obtained by the baseline method with 14,400 samples and the right column shows the trajectories obtained by our method with only 417 samples. We can see that, using the proposed critical sampling method, the network still robustly predict the long-term evolution of the system states, despite using far fewer samples for training.
Next, We experiment on the 2D Nonlinear system. In this system, we use only 925 samples to achieve similar modeling accuracy as the baseline method which has 14,400 samples. Figure I0 compares the phase plots for the 2D Nonlinear system with arbitrarily chosen initial state u(t 0) = (-1.325, 1.874) predicted by the baseline method with 14,400 samples and our method with 925 samples. The orange line represents the ground-truie -ruth solution, and the blue dotted line shows the network prediction results. In Figure 11 the left column shows the trajectories obtained by the baseline method with 14400 samples and the right column shows the trajectories obtained by our method with 925 samples.
For the chaotic Lorenz system, our proposed method only requires 1,765 samples to achieve the same modeling emorr the baseline method which requires 1,000,000 training samples. Figure 12 compares the phase plots with an arbitrarily chosen initial state u(t = 0) = (-8, 7, 27) obtained by the baseline method with 1,000,000 samples and method with 1,765 samples. In Figure [3] the left column shows the trajectories (u, U2 ux) e R obtained by the baseline method with 1,000,000 samples and the right column shows the trajectories obtained by method with 1,765 samples.
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For the last Viscous Burgers' Equation (PDE) system, proposed critical sampling method has drastically reduced the number of samples from 500,000 to 19,683 and achieved similar prediction accuracy. Figure I4 shows the predicted system states at different time obtained by the baseline method with 500,000 samples and our method with 19,683 samples, respectively. The orange line shows the ground-truth states at different time. 15 the network predictions for system states on E [0, under two different initial conditions. Both methods show accurate prediction performance under those initial conditions, but our method requires much fewer samples.
Baseline Method with 14400 Samples
Our Method with 417 Samples
Prediction
Prediction
Ground Truth
Ground Truth
2-
S
-2.0
-1.
-1.0
-0.
0.
1.0
1.
-20
-15
-2.0
-2.
8.0
10
15
U1
Figure 8: Phase plots for the Damped Pendulum Equation with initial state u(t 0)
(-1.193,-33) -.8876). Left: baseline method with 14400 samples; Right: our method with 417 sam- ples.
Baseline Method with 14400 Samples
Our Method with 417 Samples
15
Prediction
1.5
Prediction Ground Truth
10
Ground Truth m
1.0
00
00
-15
-15
-20
0.0
10.0
Prediction Ground Truth
Prediction
Ground Truth
S
-2
0.
25
3.0
75
10.0
12.5
15.0
17.5
20.0
0.
25
5.0
10.0
12.5
15.0
17.5
20.0
Figure 9: Trajectories for the Damped Pendulum Equation with initial state u(t 0) = (-1.193, -3.766). Left column: baseline method with 14400 samples; Right column: our method with 417 samples.
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Under review as a conference paper at ICLR 2023
Baseline Method with 14400 Samples
Our Method with 925 Samples
Prediction
Prediction
Ground Truth
Ground Truth
15
10
U2
05
80
00
005
-10
-10
8.0
10
-10
-05
00
05
19
U1
Figure 10: Phase plots for the 2D Nonlinear Equation with initial state u(t = 0) = (-1.325, 1.874). Left: baseline method with 14400 samples; Right: our method with 925 samples.
Baseline Method with 14400 Samples
Our Method with 925 Samples
10
Prediction Ground Truth
Prediction Ground Truth
85
0.5
5'
-0.5
-1.0
-1.0
75
15.0
17.5
2.0
20
Prediction Ground Truth
Prediction Ground Truth
S
05
-0.5
-05
-10
25
75
12.5
17.5
Figure 11: Trajectories for the 2D Nonlinear Equation with initial state u(t 0) (-1.325, 1.874). Left column: baseline method with 14400 samples; Right column: our method with 925 samples.
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Baseline Method with 1000000 Samples
Our Method with 1765 Samples
Prediction Ground Truth
Prediction
Ground Truth
20
10
0.2
-10
10 -20
Figure 12: Phase plots for the Lorenz System with initial state u(t = 0) = (-8,7,27) Left: baseline method with 1000000 samples; Right: our method with 1,765 samples.
Baseline Method with 1000000 Samples
Our Method with 1765 Samples
Prediction Ground Truth
Prediction Ground Truth
0-
-5-
-5-
Prediction
Prediction
Ground Truth
Ground Truth
5
-10
-10
-20
-20
Prediction Ground Truth
Prediction Ground Truth
5
25
20
35
15
Figure 13: Trajectories for the Lorenz System with initial state u(l 0) 7,8,7, 27). Left column: baseline method with 1000000 samples; Right column: our method with 1,765 samples.
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Our Mothoaaa with 19985 Sample Baccie Medial with 500000 Sample
104
204
304
404
432
682
425
42
470
Figure 14: Viscous Burgers' Equation: comparison of the ground-truth solution and the learned network prediction at different time. Top row: baseline method with 500000 samples; Bottom row: Our method with 19683 samples.
Forward Prodiction
Forward Prediction
Ground Troth
Baseline Method with 500000 Samples
Our Method with 19683 Samples
Figure 15:
Top row: Comparison of ground-truth solution, baseline prediction and proposed method prediction on t e [0, 2) under initial condition = 0.3770 sin(z) 0.169 sin(2x) 0.175 sin sin (3x) of Viscous Burgers' Equation system. Bottom row: Comparison
ground-truth solution, baseline prediction and proposed method prediction on E [0, 2] under initial condition u = - 00.430 sin sin(x)- (x)-
0.219 sin(2x) + 0.017 sin) of Viscous Burgers' Equation system.
24


Reviews:
[{"review_id": "PwqeBUBHb4", "reviewer": "Reviewer_C4XT", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper presents an approach to predicting a dynamical system with a spatio-temporal neural network and an active learning method for choosing a limited number of training samples to collect. They are able to greatly reduce the number of samples needed to predict a dynamical system. ", "summary_of_the_review": "I think that this paper is interesting (important problem and novel approach). However, the writing needs quite a bit of work, and I have some clarification questions that would help me understand how strong the claims are. As mentioned above, I would also like to see errors averaged over more trajectories that were not used for training. ", "strength_and_weaknesses": "Strengths:\n\nI think the problem of strategically choosing a limited training set for dynamical systems is important, and I'm glad to see more work done in this direction. The approach in this paper seems novel and creative to me. \n\nIf I understand correctly, the reported errors are on very long trajectories, whereas the temporal evolution model is trained on just one step. Long-term prediction of dynamical systems is difficult, so it's impressive to see comparisons with long-term errors. \n\nWeaknesses:\n\nI think that the writing could be improved in terms of clarity, reproducibility, and the writing on related work. (I have more details on that below.) Also, if I understand correctly, the errors reported are on 1-5 trajectories, which is very low in my experience. I would like to see errors averaged across a larger number of trajectories to see if the network generalizes well. I have additional comments in the next box.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "B-tEDS4W1e", "reviewer": "Reviewer_6jX7", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper studies the problem of critical sampling for a dynamical system by sampling from the high error regions of the state space. Since the underlying system and hence the error is not known in the region where data does not exist, the authors propose to use a surrogate error given by the reciprocal prediction error of the model. The model effective consists of two parts, a forward prediction module to predict $x_{t+1}$ from $x_t$ and a backward module to predict $x_t$ from $x_{t+1}$. The model is trained both from samples and also a reciprocal error measuring how well the prediction is going forward using the forward module for $K$ time steps and then going backward using the backward module for $K$ time steps. Some theoretical results and empirical simulation results are shown for ODEs and PDEs showing that the proposed model outperforms the baseline.", "summary_of_the_review": "I am not convinced about the motivations behind developing this technique, whether the proposed methods can be used in real physical scenarios, and conditions under which this method can be useful and outperform other models. ", "strength_and_weaknesses": "**Major Weaknesses**\n* The literature review is incomplete, many other methods for learning ODEs are not mentioned. Examples are HMMs, and linear and nonlinear state space models (LDS, SLDS, fLDS, SVAE, RNNs, etc.).\n* The original motivation for the presented method is that the existing models require large sample sizes to learn the dynamics and converge, however supporting figures are not accompanied. In time series and dynamical systems specifically, the critical sample size depends on many factors including the initial condition, sampling frequency, dynamical regime, etc. Further investigation is needed to address under what conditions a new method for critical sampling is helpful.\n* It is unclear how can the presented method be applied in practice for physical systems. Even if we know what the critical (high error) samples are, it's often the case that we cannot simply change the state of the system towards those samples. The results are only shown in the simulations, questioning whether the method is applicable to real physical systems or not.\n* Above the methods section, 4 contributions are elaborated. Contributions 1 and 2 are not independent and can be listed under the same number. Contribution 3 is not new, there is a large list of papers learning forward-backward models, i.e. simultaneously learning the forward and inverse models. In addition, the connections to HMM models are not explained, since the inference algorithms in HMM and LDS consist of a forward-backward loop, it's important to explore whether there is a connection. Importantly, HMM and LDS can handle noise, whereas in my understanding the presented model works in the noiseless regime (which is usually not the case for the real physical systems). Regarding contribution 4, I'm not quite convinced that this is fully addressed through empirical studies. The systems that are considered are very specific, comparisons are done against a specific baseline and many other models are not considered. The sampling frequency is not chosen according to the best practices.\n* The systems considered in this paper are all autonomous, this limits the scope of the presented models as there are many physical systems that aren't fully autonomous. In addition, the systems considered are low-dimensional. Since the high error regions are determined by sampling a grid in the state space, in the high dimensional cases it's unclear that the method scales properly.\n* The finding about the reciprocal error being correlated with the network modeling error is highly dependent on the underlying system, properties of the underlying flow field, learning algorithm, initial sample, dynamical regime, etc. It's unclear whether this finding holds for other physical systems.\n\n**Minor Issues**\n* The visualizations can be dramatically improved. The plots are Python's basic plots and the fonts are very small. The plots can be made more professionally.\n* What does the following mean in the second paragraph after the methods section?\n> The solution evolution is uniquely defined without bifurcation\n* In the first paragraph on page 3, \"effective\" should be changed to \"effectively\".\n* There is a grammar error in the following:\n> Learning the local spatial dynamics. The dynamical system may exist a highly nonlinear and complex behavior in the whole spatial domain ...\n\n## Post Rebuttal (Round 1)\nI thank the authors for their detailed responses and new additions to the paper. Below I reply to specific points raised by the authors in response to my criticisms.\n\n> Response: In the revised paper, we have included a separate section for related work (Section 2), and included a review of these statistical modeling/learning methods. It should be noted that, in this work, our focus is to study the problem of data-driven modeling of unknown dynamical systems whose behaviors are controlled by their underlying governing equations. These methods focus on statistical learning or modeling of more generic system observation data.\n\nOne interpretation of the models I mentioned such as HMMs and linear and nonlinear state space models (LDS, SLDS, fLDS, SVAE, RNNs, etc.) is that they are approximating an underlying dynamical system or ODE using a certain function family determined by the model. For example, in the case of LDS this function family is linear ($x_{t+1} = Ax_t$) whereas for SLDS this function family is piecewise linear and for RNN it can be a generic nonlinear function parameterized by a neural network $x_{t+1} = f(x_t)$. In this sense, I don't see a difference between the articulated model and these existing frameworks and I don't find the reason convincing enough to exclude a comparison with these models.\n\n\n> Response: Thanks for pointing out this. In existing work on data-driven modeling of dynamic systems, they need large sets of training samples to successfully train the deep neural network to achieve a reasonably accurate approximation of the system evolution trajectories. Table 1 of the revised paper summarizes the number of needed training samples for the method to achieve the desired model prediction error. For example, for the first Damped Pendulum system, the baseline method needs 14400 samples to achieve an average prediction error of 0.0263 so that the predicted system trajectory is fairly close to the true trajectory.\n\nAs I mentioned in my original reply, intuitively the number of samples needed depends on many factors, including but not limited to the sampling frequency, dimension, and dynamical regime (e.g. chaotic, stationary, stable, etc). Since the main motivation for this work is that models require large sample sizes, this should be illustrated more thoroughly through empirical results on multiple systems when best practices are used (e.g. optimal sampling frequency and models with lower sampling requirements). A vanilla comparison with an ad hoc method where no consideration of the best practices does not sound like a convincing baseline, to begin with.\n\n> Response: We really appreciate these insightful comments! We totally agree with you that in practical systems, it is not straightforward to change the state of the system towards those samples. Here are our thoughts and responses to address the issues that you pointed out ...\n\nI was expecting to see some results on physical systems in response to this question. If this is applicable only to simulation I don't find the contribution significant enough for publication. If this is in fact applicable to physical systems (as the authors argue it is) results must be accompanied.\n\n> Response: ... Sorry for the confusion. Our idea and method are quite different from the HMM and LDS. Specifically, in our work, we construct a multi-step reciprocal prediction network structure, which performs K-step forward prediction followed by K-step backward prediction by concatenating K forward prediction networks plus K backward prediction networks together. Interestingly, we find that this multi-step reciprocal prediction error obtained from this prediction loop is highly correlated with the model prediction error.\n\nI understand the motivation and the presented method and how they are different from LDS and HMM. I apologize for the unclear question and I hope this clarifies the question further. Regardless of the motivation of your paper, the algorithm that you've presented has forward and backward parts where you run the model forward through the forward mapping and then backward through the inverse mapping and evaluate the error and backpropagate through the parameters of the model. Indeed you use the error to efficiently sample from the dynamical system for the next iterations and hence the active learning framework. Now my question is, the forward-backward algorithm appears in other time series models as well (such as Kalman filter, LDS, HMM, etc.)? It would be interesting to see if there are connections between the two sets of algorithms.\n\n\n> Summary of Responses ...\n\nI read the responses from other reviewers as well and I appreciated their input about the work. I do think however that some of the points that I raised remain concerning and unanswered and they are independent of the points raised by other reviewers. In summary, I think the empirical results do not fully support the claims in the paper and more systematic empirical evaluation is needed for backing up the arguments. Specifically, considering other models for comparisons, using best practices as the baseline, and considering underlying dynamical systems with different characteristics (dynamical regime, dimension, presence of noise, etc.) is critical for the claims to hold. Given this, I tend to keep my score as is.\n\n\n## Post Rebuttal (Round 2)\n\nI thank the authors for further explanations, in the following I wish to clarify my standpoint and the reasoning behind my score further.\n\n**Regarding the motivation of the paper,** it is mentioned that learning ODEs using deep networks require large sample sizes and critical sampling helps with reducing the sample size. While I think that this statement is true in general when a baseline model and vanilla sampling schemes are used for comparison, I think it remains unclear under what circumstances it's worthwhile to use the proposed framework. There are ad-hoc methods for reducing the sample size. For example, one could subsample the input data using a frequency given by the auto-correlation time of the time series. Or one can run a clustering and subsample within a cluster.\n\nI think the answer to this question depends on many factors, some of which are listed below.\n\nIt depends on the underlying model. If the underlying deep network utilizes linear functions, the sample size required for learning the dynamics is as low as the dimension of the system. While the linear model is just an example, it shows that certain deep learning function spaces (determined by the neural network architecture, choice of nonlinearity, number of nodes, and regularization) will inherently require fewer data sizes to be trained compared to others. If the effect of inductive biases of the function space on the sample size is not systematically studied it is not possible for a practitioner to decide if this framework will be useful for their case or not.\n\nIt depends on the dynamic properties of the data. For example, if the data is sampled from an attractor type or stationary ODE, intuitively, fewer samples should be needed to learn the dynamics compared to a chaotic attractor (e.g. Lorenz model). It's important to show under what regimes the claims hold so that the practitioners are able to determine if the method is applicable to their case or not.\n\n**Regarding the significance of this work,** I wish to clarify that I do not think that every paper needs to have experimental results in it. I have given high scores to papers with no figures and no data before and I do not question the importance of having empirical studies on simulations and theoretical studies. However, I do believe that if a method is developed mainly for practical settings, accompanying it with real data will make it stronger. In my opinion, the methodological novelty of this paper is limited. Leaning forward and backward functions existed before. The main novelty comes from the empirical finding that the multi-step error is correlated with the ground truth error, hence utilizing it during sampling will help reduce the error faster than vanilla sampling. Given that the novelty lies in this empirical observation, at the very least a systematic investigation of the conditions under which the claim holds is required.\n\nGiven that in the simulated cases the constraints imposed by the computational resources are less severe and we can train deep learning models in a short time, the main merit of this work lies in experimental settings where we are time and budget limited. Given the context I provided, unless some results on real-world data are not shown (by that I certainly do not mean that the authors should do the experiments themselves) I do believe that the scope of this work is limited. In addition, noise is an inseparable component of real-world data, in the presence of which learning the forward and backward model might be more challenging and the critical sampling framework might not hold.\n\nIf we accept that the scope is limited, and we base the judgment only on the simulation results, then more controls are needed to show that if we account for time complexity and other factors, then the presented framework has better inductive biases and achieves better test/generalization error. Creating a grid of points and running the forward-backward model for a few time points is a time-consuming process. To bring the comparisons to the same footing, one should consider the time spent on the optimization and not just the number of samples used for training. Additionally, deep learning models are proven useful mainly in high-dimensional settings. There is no systematic study of how much of the claims hold in higher dimensions and the feasibility of applying the method to higher dimensional settings in case of evaluating the error on a grid and training the forward and backward functions. I think that for the acceptance of an empirical paper in ICLR, a solid empirical investigation (including the ones I mentioned above) is necessary.\n\n\n\n## Post Rebuttal (Round 3)\n\n> Response: Thanks for pointing out this! It should be noted that, in this paper, we are learning deep neural network models of unknown dynamical systems ...\n\nThe auto-correlation is easily estimated from the data, it is the correlation of a signal with a delayed copy of itself as a function of delay. There is no need for the equations to be known. Regarding clustering, one can think of clustering as a way to sparsify the data samples. Start from a large sample size, run a simple clustering algorithm such as k-means and use the cluster centroids or data points that are close to cluster centroids for training a neural network. These are the simplest ad-hoc algorithms to reduce the data size, but more complex ones can also be considered. My point is, in your work you have a data selection process, which is to evaluate the error on a grid of samples and choose the high-error data points for training. This data selection process should be compared to other data selection processes such as the ones I mentioned above as opposed to the vanilla fixed-interval sampling scheme used in the comparisons to motivate the paper.\n\n\n> Response: Thanks for this insightful comment! We totally agree with you that the number of needed training samples does depend ...\n\n> Response: We agree with you that experiments on real-world data will be very helpful. We will follow your comment to ...\n\nI cannot base my judgment on the results that I have not seen. Once the new results are included I will re-assess the paper accordingly.\n\n\n> Response: We totally agree with you that learning different ODEs would require different amounts of data. In our problem, ...\n\nThis needs to be carefully evaluated and investigated. I understand that in the problem statement you are assuming that the governing equations are not known. But ground truth studies can investigate under what dynamical regimes (with known equations) the method performs better than others. \n\n> Response: Thanks for pointing out this! In this paper, we study the important problem of selecting ...\n\nGiven that in the simulated environment the proposed method takes longer to run compared to baseline methods, now I firmly believe that the main merit of the paper is in the real-world scenarios where data collection is challenging and costly. In the simulated setting, one is not concerned with the computational resources of using a larger data sample for training. In fact, this just shows another example of the no-free-lunch-theorem. You are using the computational resources in a different way (computing multi-step error for a grid of samples) compared to using the computational resources for processing batches of data.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "6TN68uQAym", "reviewer": "Reviewer_mGbk", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper studied an interesting and important problem and tried to answer the question: given an unknown dynamical system, how to predict its evolution behavior accurately with minimum number of samples and how to select these samples. To be specific, the paper introduces a multi-step reciprocal prediction network, where a forward evolution network and a backward evolution network are designed to learn and predict the temporal evolution behavior in the forward and backward time directions, respectively. In addition to that, the paper also proposed a joint spatial-temporal evolution network which extends the framework to the space. The numerical result is appealing and demonstrates promising performance compared to baseline approach. \n", "summary_of_the_review": "See above", "strength_and_weaknesses": "This is an interesting paper, which studied an important problem in dynamical system modeling. The methodology is well presented and easy to follow. I also want to commend on the numerical results and their figures, which are easy to be interpreted and demonstrate promising performance. The finding of strong correlation between the network modeling error and the multi-step reciprocal prediction error is super interesting and it is also very smart to use the multi-step reciprocal prediction error to approximate the desired network modeling error. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject