A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments
Authors: Yuchen Wang
Hongjue Zhao
Haohong Lin
Enze Xu
Lifang He
Huajie Shao
Venue: ICML 2025 poster (2025)
Keywords: Physics-enhanced Machine Learning
State Space Model
Long-term Dynamics Forecasting
Dynamical Systems
Primary area: deep_learning->sequential_models_time_series
TL;DR: We propose Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting.

Abstract: This work aims to address the problem of long-term dynamic forecasting in complex environments where data are noisy and irregularly sampled. While recent studies have introduced some methods to improve prediction performance, these approaches still face a significant challenge in handling long-term extrapolation tasks under such complex scenarios. To overcome this challenge, we propose Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting in complex environments. Our motivation is that SSMs can effectively capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge improves generalization ability. The key challenge lies in how to seamlessly incorporate partially known physics into SSMs. To achieve this, we decompose partially known system dynamics into known and unknown state matrices, which are integrated into a Phy-SSM unit. To further enhance long-term prediction performance, we introduce a physics state regularization term to make the estimated latent states align with system dynamics. Besides, we theoretically analyze the uniqueness of the solutions for our method. Extensive experiments on three real-world applications, including vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting, demonstrate the superior performance of Phy-SSM over the baselines in both long-term interpolation and extrapolation tasks. The source code will be publicly available upon publication.

A Generalizable Physics-Enhanced State Space Model for
Long-Term Dynamics Forecasting in Complex Environments
Yuchen Wang Hongjue Zhao?
Haohong Lin
Enze Xu
Lifang He
Huajie Shao
Abstract
1. Introduction
This work aims to address the problem of long- term dynamic forecasting in complex environ- ments where data are noisy and irregularly sam- pled. While recent studies have introduced some methods to improve prediction performance, these approaches still face significant challenge han- dling long-term extrapolation tasks under such complex scenarios. To overcome this challenge, propose Phy-SSM, a generalizable method that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting in complex environments. Our moti- vation is that SSMs can effectively capture long- range dependencies in sequential data and model continuous dynamical systems, while the incor- poration of physies knowledge improves general- ization ability. The key challenge lies in how to seamlessly incorporate partially known physics into SSMs. To achieve this, we decompose par- tially known system dynamics into known and unknown state matrices, which are integrated into Phy-SSM unit. To further enhance long-term prediction performance, state regularization term make the estimated latent states align with system dynamics. Be- sides, we theoretically analyze the uniqueness of the solutions for our method. Extensive ex- periments on three real-world applications, in- cluding vehicle motion prediction, drone state prediction, and COVID-19 epidemiology fore- casting, demonstrate the superior performance of Phy-SSM over the baselines in both long- term interpolation and extrapolation tasks. The code is available at https: : thw..........e.g...... com/ 511205787 PPM ICML2025.
Dynamical systems have been widely applied across a broad range of real-world domains, including autonomous driving (Kong et al., 2015; Rajamani, 201 I), epidemiology (Nicho, 2010), and climate science (Ionides et al., 2006).
Generally, dynamical systems are often governed by un- derlying physical laws. Motivated by this, some studies have developed physics-enhanced machine learning mod- els (O'Driscoll et al., 2019; Cicirello, 2024) for to enhance the generalization ability of dynamics forecasting by incor- porating known physical laws, such as energy conservation and differential equations (Greydanus et al., 2019; Raissi et al., 2019). A common assumption in these methods is that the physical laws governing system dynamics are fully known as prior knowledge. However, in practice, it is chal- lenging obtain the complete governing equations for com- plex dynamical systems using first principles (Linial et al., 2021; Gentine al., 2018). On the other hand, dynamical systems like autonomous vehicles operate in complex and unknown environments such as inclement weather condi- tions, their sensors often suffer from faults or mismatched clocks (Dabrowski & Rahman, 2019; Huang al., 2023), resulting in noisy and irregularly sampled data.
introduce physies
To address these issues, few recent works (Linial et al., 2021; Takeishi & Kalousis, 2021; Yang al., 2022) have devel- oped partially known eehysccee machine learning models for noisy, regular data. While these methods per- form well in interpolation tasks, they often struggle with long-term extrapolation tasks with irregular data. An illus- trative example provided in Appendix A. This limitation arises from their solutions heavily relying initial condi- tions, lacking effective mechanism to dynamically refine predictions based on subsequent observations (Chen et al., 2024; Kidger et al., 2020). Thus, the question is: how to en- hance accuracy and generalization for long-term dynamics forecasting with noisy, irregular data?
'iiliiam& Mary University of Illinois Urbana-Chaan Champaign e Carnegie Mellon University "Lchigh University. Correspondence to: Huajic Shao <hshao@ wm.edu>.
In this work, we propose Phy-SSM, generalizable method that incorporates partially known physics into deep state- space models (SSMs), as shown in Fig. 1. Our motivation is that deep SSMs (Gu et al., 2021; Smith et al.; Gu & Dao, 2023) can not only effectively capture long-range dependen-
Proceedings of the = International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Physics-Enine Enhanced State Space Model for Leon-TTemm Dynamics Forecasting
(t..)
Inputs
Phy-SSM Unit
Sequential Encoder Layer
u u(aa)
u u(fm-1)
(1,)
SSM
No
Ann
Discret
((4)
2(f1)
x(tn-1
314r)
layer
Latent States
Phy-SSM
...
Phy-SSM
Fusion
MA
A
(ta)
(t.
(fi+1)
M SSM
B
B
Discrct
Decoder
layer
M
Outputs
a=ff))
(a) Overall architecture of Phy-SSM
(b) Architecture of Phy-SSM unit
Figure 1. (a) The overall architecture of Phy-SSM, consisting of three components: sequential encoder, Phy-SSM Unit, and a decoder. (b) Detailed architecture of Phy-SSM unit.
cies in sequential data but also model continuous dynamical systems. However, developing Phy-SSM involves address- ing two main challenges: (i) seamlessly integrating partial physics knowledge into the model architecture, and (ii) fa- cilitating long-term predictions in the presence of noisy and irregularly sampled data. To tackle the first challenge, we develop novel Phy-SSM unit that decomposes partially known dynamics into known and unknown state matrices, as shown in Fig. 1(b). This decomposition represents sig- nificant advancement compared to existing deep SSMs (Gu & Dao. 2023; Gu et al., 2022). For the second challenge, we introduce a physics state regularization term to further constrain the estimated latent states by the encoder to com- ply with the system dynamics. Furthermore, we provide theoretical analysis of the uniqueness of solutions for modeling partially known dynamical systems. Finally, we evaluate the proposed Phy-SSM on three real-world applica- tions: vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting. The results demon- strate that our method significantly outperforms baseline methods for long-term interpolation and extrapolation tasks. These findings highlight the effectiveness incorporating partial physical knowledge into deep SSMs for improving predictive generalization complex, woll-wrlld scenarios.
that Phy-SSM significantly outperforms baseline methods in long-term dynamics forecasting.
2. Related Work
Machine Learning (PEML). Depend-
ing on how underlying physics knowledge is incorporated into models, PEML. (Faroughi et al., 2022; O' O'Driscoll et al., 2019; Cicirello, 2024) can be classified into two main types:
(i) Physics- Inforee- Loss Function. This method incorpo- rates physical laws into the loss function as soft constraint, ensuring that ML models remain consistent with the laws of physics (Baydin et al., 2018; Chen et al., 2020; Wang et al., 2023; Raissi et al., 2020; Yu et al., 2022: Raissi et al., 2019). typical line of this method PINN (Lu al., 2021; Raissi al., 2019) that integrates differential equa- tions into the loss. However, such methods often struggle to extrapolate beyond the training distribution (Bonfanti et al., 2024; Kim et al., 2021), since they are trained to conform to the solutions within a pre-specified domain.
(ii) Physics-Informee Architecture Design. This approach tries to embed physical principles into the design of ML architectures enhance model generalization. Some works focused on neural ordinary differential equations (NODEs) (Chen et al., 2018) that adopted deep neural net- works (DNNs) to parameterize underlying ODEs. For in- stance, a recent study developed ContiFormer (Chen et al., 2024) that combines Transformer with NODEs to model continuous-ting dynamics on irregular time series. This method, however, very computationally expensive and may struggle to extract generalized physical representa- tions for long-term predictions. Other studies designed
1) We propose Phy-SSM, novel approach that integrates partially known physics into state-space models to improve generalization for long- term forecasting in complex environments: 2) To enhance long-term prediction accuracy, we introduce physics state regularization term that constrains latent states align with system dynamics; 3) We offer theoretical analysis of uniqueness of solutions for our method; and 4) Extensive experiments on three real-worke applications demonstrate
Our contributions are four-fold:
2

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
ML to model energy-conserving systems by complying with Hamiltonian mechanics (Greydanus et al., 2019; Bacsa et al., 2023; Zhong et al.) or Lagrangian mechanies (Cran- mer et al., 2020; Lutter et al., 2018). While these methods improve model generalization ability due to their embedded physics inductive bias, their reliance on specific physics knowledge limits their broader applicability as a general- purpose approach.
fixed emission function g:
(x1) - 9 (z()),
where x: [0, + X C R. and 2 : [0, + C Rd, represent the observations and full system states, respec- tively. In practice, the system states (t) are hardly observ- able, governed by the underlying system dynamics below:
dz(t) = f (z(1), u(t),
(2)
In summary, these existing methods did not consider real- world dynamical systems where obtaining complete physics knowledge is often infeasible (Linial et al., 2021).
dt
where ual)) represents the control input that influences the system, and f denotes governing equation of system dy- namics, which is often partially known.
Partially Known Physics-Based ML. Some works have focused on partially known physics- machine learning (ML). For instance, SINDy Autoencoders (Champion et al., 2019) identified physical laws directly from data by opti- mizing a linear combination of predefined functions. Phy- Taylor (Mao et al., 2023) integrated system dynamics into its architecture design. However, these approaches rely on finite difference techniques to estimate derivatives, making them only applicable to noise-free and regular data (Cham- pion et al., 2019; Yin et al., 2021).
Our Problem. The goal of this work is to predict the long-term trajectories of a dynamical system in complex environments where data are noisy and irregularly sampled. Specifically, the inputs include the observational sequence ax> (fo)), ax(1); :..)).......)... (tm))] with n data points, where 0
< 4 sequence [uu(la). (((()).).,,,(()))))) , (((m))),,,,(+)))))) where I is the length of the prediction window.
< in T. and the control input
Our tasks will involve: i) Interpolation: Predict the se- quence (m()), ((1),..... (((1))).,,((()))) using the learned dynam- ies. Here, æ(t:) is the predicted observation t,, obtained using our method. ii) Extrapolation: Forecast the future trajectories (1) for L E (lm:
Recently, few studies (Yin et al., 2021; Linial et al., 2021; Wehenkel et al., 2023; Takeish & Kalousis, 2021) have explored partially known physics-enhanced machine learn- ing that use physics-based NODES to model underlying dynamics. While these methods have proven effective for continuous-time modeling, their reliance on NODEs presents challenges in accurately capturing nonlinear and time-variant systems over long time horizons. This limi- tation arises from NODEs' heavy reliance initial con- ditions, which hinders their ability to capture subsequent long-term sequence correlations. Additionally, these meth- ods are only employed to regular data rather than irregular data in the experiments.
3.2. Structured State Space Model
We introduce the foundation of Structured State Space Mod- els (S4) (Gu et al., 2021), which serves as the core compo- nent of the proposed Phy-SSM. S4 is an emerging neural architecture built on classical state space models (SSMs) from control theory, designed to capture long-range depen- dencies in sequential data. To better understand $4, we first review the basic knowledge of classical SSMs as follows:
Different from prior work, we propose generalizable model that leverages partial physics knowledge to enhance the generalization of long-term dynamics forecasting with noisy, irregular data.
(t) Az + Bu(t),
(3)
yt)) = Cz( + Du((),
where A E RRxx, B E R=xm C RRX" and D ppxm are the state, input, output, and feedthrough matrices,
3. Preliminaries
respectively. In addition, u(t) E Rm represents the input signal, xtt) R" denotes the state variables, and yy()) R represents the outputs. Note that Du(() = 0, as Du(t) can be interpreted as a skip connection.
Notations. The detailed descriptions of important notations are presented in Table in Appendix B.
3.1. Problem Statement
For practical applications involving discrete sequences, the continuous model needs to be discretized. Com- mon discretization methods, such as bilinear transforma- tions (Tustin, 1947), can be employed. By discretizing with step size A, the system is expressed by the following linear recurrence relations:
We first review the background of dynamical systems and then describe our research problem.
Background. Given a set of observations xtt) from dy- namic system. We assume that the observed trajectory is governed by the underlying system dynamics z through a
Zn = Az 1 + Bun
(4)
yn = Czni
3

---

Physics-Enhaneed State Space Model for Long-Term Dynamics Forecasting
where A, B, and C are the discrete-time parameters derived from A, B, and C' with the step size A. These parameters preserve the dynamics of the continuous model in the dis- cretized setting, enabling effective modeling of sequential data. S4 (Gu et al., 2021) extends the use of SSMs for modeling long sequences. In particular, S4 leverages a spe- cialized matrix initialization technique called HiPPO (Gu et al., 2020) to efficiently maintain information from past inputs. Building such as S5 (Smith et al.) and Mamba (Gu & Dao, 2023) have been proposed in recent years.
and jú (4), (1) respectively represent the mean and vari- ance of the learned posterior distribution. The process is stochastically approximated using the reparameterization trick (Kingma, 2013).
Next, we need to approximate the posterior distribution (==((gg) x((gn)) based on the above Eq. (5). To reduce computational costs, inspired by prior work (Girin et al., 2020), the posterior distribution can be simplified as
S4, multiple variants of deep SSMs
q(=(t<n) x((=n))
II (ttg)))
(6)
Among these, recent work (Smith et al.) introduced a S5 model that employs parallel scans to accelerate training in a recurrent mode. This design allows the model to efficiently handle time-varying SSMs. Its key advantages include: (i) It can capture long-term data dependencies through the formulation
Phy-SSM Unit. The primary role of the Phy-SSM Unit is to enforce known physical laws while simultaneously learning unknown dynamics from sequential data. Assuming that the latent system states conform to certain physical dynamics, the Phy-SSM Unit leverages the latent states and control input from the previous time step as input for generating physics- chysiss-consistent predictions. However, developing the Phy-SSM unit poses two key challenges: 1) incorporat- ing partial physics knowledge into model design, and 2) accurately modeling the unknown dynamics for long-term predictions.
HiPPO memory, and (ii) Its continuous- enables the effective modeling of irregularly sampled data. Inspired by these, we developed a physies-enhanced SSM that incorporates partial physics knowledge for long-term dynamics forecasting in complex environments.
4. Proposed Method
4.1. Framework of Phy-SSM
To address the first challenge, we enforce known physies knowledge through dynamics decomposition. Specifically, consider the system dynamics in Eq. (2), we decompose it into known and unknown parts as follows:
As mentioned in Sec. 3.1, our goal is to enhance long-term predictions of dynamical systems when data are noisy and irregularly sampled. To achieve this, we propose a novel Phy-SSM that incorporates partially known physics knowl- edge into the model design, shown in Fig. 1. Phy-SSM is composed of three main components: sequential encoder layer, the Phy-SSM Unit, and decoder, with the Phy-SSM Unit serving as the core. The key idea behind Phy-SSM is to first employ a sequential encoder to encode the inputs to approximate the posterior distribution of the latent system state. Next, the Phy-SSM Unit jointly takes in the past la- tent state and control inputs to predict the next latent states. Finally, the latent states generated by the Phy-SSM Unit are fed into the decoder to produce the final output. Below, we elaborate on each component in the proposed framework.
dz(() f (z(t), u() dt
= funw (z(t).u(t)) + funk (z(t).u(t))
Inspired by prior work (Brunton et al., 2016), we transform the system dynamics into a linear SSM by extending the state to a new state 2, which includes z and can addition- ally incorporate nonlinear terms or constants. This method can help to represent nonlinear systems in linear manner. By doing this, it enables us to: (i) use matrix calculation to improve computing efficiency, and (ii) embed knowledge into matrices directly.
Based on this motivation, we convert the above Eq. (7) into the following linear SSM formula.
Encoder for Posterior Probability Estimation. We first adopt a sequential encoder & to estimate the posterior distri- bution of the latent system states zt)) based on observations at).. The sequential encoder is simplified structured SSM that can handle irregular data while introducing a memory variable htt) to capture long sequence correlations. Specif- ically, the approximate posterior 2(t;) at each time step depends not only on (tt,) but also on the memory from the previous time step, (t.-1)). Mathematically, we have
dz(())
di
(8)
(Axaw (t) + Aunk (t))=(t) + (tnu(t)
where A is the state matrix, and B is the input matrix. The extended state is defined as Σ = [z (T E Rd= where (b(2) E RRde represents additional extended terms. Here, ARnw E Rddxdds Aunk E &dd and Bunk E dd xd_ Specifically, A denotes the known phys- ical dynamics, while Aunk (t)((t)(()))))))) represents the unknown system dynamics. Bunte()) (t)u(t) models the influence of con- trol inputs. In Eq. (8), we omit Bro since the influence
=(t,) (xt)) ~ N' (t.),,), (iag(o?((t))))
where
(e(t)),oz(t)) = sp
4

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
of control inputs is often unknown; otherwise, it can be treated in the similar way as Akow- After decomposition, it becomes straightforward to enforce explicit physical knowl- edge into the model. A detailed example of illustrating this transformation process is provided in Sec. 4.3.
standard Gaussian distribution, we directly embed physies knowledge into latent space. Consequently, the prior probability of the latent =(t) is defined as:
x(t)) ~ N (Ha(tt), dit)(
where (t,)t) and (t.) represent the mean and variance of the physics-based prior distribution. To respect the physical dynamics, we use the Phy-SSM Unit to generate μ4g (4: and (t;)). Specifically, the outputs of the Phy-SSM unit are passed through a linear map to compute μt (t:) and (t.).
To address the second challenge, inspired by the powerful HiPPO memory mechanism (Gu et al., 2020), we utilize multi-layer structured SSMs model continuous unknown dynamics. The structured SSMs enable the Phy-SSM unit to memorize long-term historical patterns and accurately estimate unknown dynamics.
During the interpolation stage, the continuous Phy-SSM unit dynamically refines its predictions using the posterior from the preceding time step within the observation window, This approach effectively reduces prediction error stem- ming from potentially inaccurate initial conditions. During the extrapolation stage, Phy-SSM leverages accurate initial conditions and well-learned physics to perform predictions through autoregression. Once the full trajectory of physics latent states obtained, according to (Girin et al., 2020), the decoder maps these latent states to the output, yielding
Based on the above ideas, we present the detailed structure of the Phy-SSM Unit, as illustrated in Fig. ((b). Specifically, the Phy-SSM Unit involves the following three key steps for physics-based latent state prediction.
1) Learning unknown continuous functions A (l) and B Bunk (t) using structured SSMs. Multi-layer struc- urrdd SSMs take and u as input to approximate the unknown continuous functions Aunk &, u; 0A) and Bunk (t, 2, u: BB) The outputs of the structured SSMs are passed through a fully connected layer and reshaped into Rdixd;
p(x((ts.), x(tr,)) = I[(((()) ((t))((z(() ((t-)))).
(11)
2) Knowledge mask to encode physics as hard constraints. To constrain the model to learn only unknown terms, we implement& a simple yet effective knowledge mask mecha- nism. Specifically, we introduce a binary knowledge mask M (0. d2 x that is applied via the Hadamard product to the learned unknown terms. The positions with value of 1 in the mask indicate that the corresponding dynamic item is permitted to be updated, while positions with value of 0 block the influence of the item. The learned unknown dynamics are then refined as follows:
Overall Objective. The objective function comprises the the negative time step-wise variational lower bound (Chung et al., 2015) and the physics state regularization term below:
LVAE + >fregg
(12)
where
n LVVE > E (( Cin - -BckLl
(() logp(x((t,) (t())),
recon
Aunk (f) = MA O Aunk(!),
Lk = KT. (g(2(()))) (((ss))) |l p(=(t)) (x((-)))),
(9)
Bunk (t) = Mn @ (t).
Luz
|1=(tt) - (t:))2
For detailed knowledge mask design for different dynamical systems, we provide a guideline in Appendix I.
T2
Here, the first term Creeon in LVAE represents the recon- struction loss, capturing how well the model reconstructs the observations. The second term, LKL. quantifies the Kullback-Leibler (KL) divergence between the prior and posterior distributions of the latent states. Creg represents regularization term. =(t,) and x" (t:) are latent states sam- pled from the prior and posterior distributions, respectively. The hyperparameters 3 and X control the trade-off among the reconstruction loss, KL divergence, and regularization terms in the loss function.
3) Discretizing continuous dynamics to compute the next latent state. After obtaining Akow (t) and Aunlet)) we can compute the full system dynamics matrix A in Eq. (8). To discretize the continuous-tine model for generating latent states, we use the bilinear method (Tustin, 1947), which converts the continuous state matrix A into its discrete ap- proximation A. Notably, our model retains the continuous parameter A, enabling it to handle irregularly sampled data. After discretization, the Phy-SSM unit generates the next time-step output:
Note that the physics state regularization term Grog serves two purposes. First, it constrains the output of the sequen- tial encoder to adhere to the physical dynamics. Second, it facilitates the Phy-SSM unit learn more accurate unknown
E((+11) = (()) + (()))
(10)
Prior Physics State Prediction for Decoder. Unlike standard VAE, where latent variables typically follow a
5

---

Physics-Enine Eensnns-Ennaneed State Space Model for Long-Term Dynamics Forecasting
dynamies that align with the entire trajectory, improving performance in extrapolation tasks. In practice, this term is implemented as Euclidean distance penalty between the sample (t;)) from the prior distribution and the sam- ple =1,),) from the posterior distribution. The choice of Euclidean distance as the regularization metric is based on empirical evaluation. Detailed comparisons with alterna- tive metries are provided in Appendix The effectiveness of this regularization term is validated through following experimental results and ablation studies.
corresponding experimental results are presented in Ap- pendix F.
Based on the partially known system dynamics, we can perform state augmentation by extending the nonlinear state terms, such that s(t) = sin (f(t)) and c(t) = cos (f())).
Thus, the above Eq. (13) can be rewritten as the following stace-space model:
TO())
0
1
TO(C)
dt
d
(()
s(t)
=
10000
b
w((t)
w(t)
s(t)
: (14)
c(tt]
-w(1
c(t)
4.2. Theoretical Analysis for Dynamics Decomposition
where -4 is the unknown term, and I represents the un- m
We further offer theoretical analysis of the proposed dy- namics decomposition, illustrating the uniqueness of the solutions during model learning. Given a dynamical system with partially known terms and parameters, we have the following proposition.
known parameter for the pendulum length.
Then, AKnW (1) and Aunk (f) are denoted by:
0 1
0
0
Azow (t)
0
0
(()
Proposition (Uniqueness). For a dynamical system in the form of Eq. (7), if it can be reformulated as Eq. (8), the decomposition in Eq. (7) that minimizes Eq. (12) is unique.
LO
0
(w(
0
Au)k(()
* *
*
0 0
0
The key insight is that Akow and Aunk have disjoint support; i.e., no overlapping entry is used by both matrices, ensur- ing they do not interfere with each other during training. Detailed proofs are provided in Appendix C.
0
0
0
where the first, third, and fourth rows of Auuk(t) are trivial and thus set to zero. The second contains (+), which represents unknown parameters or terms dependent on the state and control inputs.
4.3. A Walk- Walk-Through Example
Next, we introduce the three steps of the Phy-SSM Unit for physics state prediction in our pendulum example:
In this section, we present illustrative example of video pendulum to aid in understanding the main pipeline of the Phy-SSM unit. Consider a series of videos where the un- derlying physics corresponds to a pendulum with unknown friction. The dynamics of the pendulum are governed by the following differential equations:
1) Learning unknown continuous functions Aunie(t) using structured SSMs. For the pendulum case, the output of the structured SSMs is a x 4 matrix Aunk (t) containing unknown elements (+), similar to Aunk-
2) Knowledge mask to encode physics as hard constraints. In this step, we apply a knowledge mask through the Hadamard product with the learned unknown dynamics to encode physics as hard constraints. In our pendulum example, the knowledge mask is defined as
dd(t)) df
= wt),
(t) dt
b
(13)
sin ((t)
(t),
unknown
unknown
Aunk (1) = MA @ A (t),
where ((t) represents the angular displacement of the pendu- lum, (t) denotes its angular velocity, g is the gravitational acceleration, l is the length of the pendulum, b represents the damping coefficient, and m is the mass.
0
0
0
0
1
1
1
(15)
where MA
0
0
0
0
0
0
0
0
In real-world applications, it is difficult to obtain the com- plete system dynamics. However, it is not very hard to ob- tain part of system dynamics based on domain knowledge. In this example, the pendulum length I and the damping force caused by friction, in (!), are unknown. For sim- plification, the control input to the system is omitted. The unknown impact of the control input B Bunk can be handled in the same manner as Aunk- A full system description, including Bunk, is provided in Appendix D.2.4, and the
3) Discretizing the continuous dynamics to generate the next latent state. After the second step, we obtain the continu- ous matrices Axow (t) and Aunk (t), which are combined to compute the full dynamics matrix A. Next, we apply the bilinear method to convert A in Eq. (8) into its discrete approximation A, enabling the next time-step prediction as described in Eq. (10).
6

---

Physics-Enhaneee State Space Model for Long-Term Dynamics Forecasting
Table 1. Performance comparison of different methods in terms of interpolation and extrapolation using drone dataset. The data high-fregge frequency and irregularly sampled, recorded at nearly 1010 Hz (minimum: 573.05 Hz, maximum: 1915.86 Hz). The results are averaged over three random seeds. The lower is the better.
Interpolation Task
Extrapolation Task
Method
MAE J. (x10-l)
MSE 4 (x10-l)
MAB ( (x10-1)
MSE. (x 10-
Latent ODE (RNN Enc.)
3.180+0.069
2.584+0. 117
3.610+0.041
3.425+00.14
Latent ODE (ODE-RNN Enc.)
3.304400066
2.779 142
4.437:0.212
5.551=::421
ContiFormer
128
0.004
4.059+0.024
5.092:+0. 138
S5
1. 0.15
0,309=40,076
8,426=1.351
17.333+4.855
GOKU
3.293=003377
2.738:+0.590
3,456=0.289
3.130=#.5778
PI-VAE
3.006
100016
3.627
3,58910.011
SDVAE
3,701=0.100 103
3.542++.221
3,808+00098
3.921+0.199
ODE2VAE
3.412.10.012
2.942-10.024
3.46110.012
3.115:10.020
Ours
1.00:::0034
0.222:10.020
2.733110059
1.798
Table Performance comparison of different methods in terms of interpolation and extrapolation using Covid-19 dataset. The data contains 10% missing daily records. The results are averaged over three random sceds. The lower is the better.
Interpolation Task
Extrapolation Task
Method
MAE (x
MSE | (x10-2)
M (x1001)
MSE (x 10-1)
Latent ODE (RNN Enc.)
1.14810.047
2.642.10.205
6.60510.175
8.37010.616
Latent ODE (ODE-RNN Enc.)
0.991=0.097
1.983=00410
6.846=0.295
9.304 1.322
ContiFormer
0.83011..156
1.05910.390
6.882100158
9.147.10.337
S5
0.86110.151
1.057.1..255
5,212110.554
4.560.10.177
GOKU
1.019+0.138
1.667+0. 157
6.140+0.651
7.918++0.653
PI-VAE
1.18610.353
2.45411.186
6.29211.263
8.775
SDVAE
2.290++0.212
7.908 +108+91.599 1,594
6.811
9.584+0.473
ODE2VAE
0.390
4.008
7.420++0.724
8.051
Ours
0.795=0.208
1.02++0.538
1.998+0.753
0.692++0.486
5. Experiments
3) Symplectic DVAE (Bacsa et al., 2023), 4) ODE2VAE (Yildiz al., 2019)
We first conduct extensive experiments to evaluate the per- formance of Phy-SSM using three real-world applications. Then, we conduct ablation studies to explore the impact of
For vehicle motion prediction, in addition to the above physics-enhanced baselines, we include three additional SOTA data-driven methods specifically designed for this task: 1) Wayformer (Nayakanti et al., 2023), 2) AutoBot (Girgis et al., 2022), and 3) G2LTraj (Zhang et al., 2024).
key components on model performance. The experimental settings and system dynamies are detailed in Appendix D.
Real- World Datasets. We evaluate the performance of the proposed method on three real-world applications with irregularly sampled data: drone state prediction (Eschmann et al., 2024), COVID-19 epidemiology forecasting (Takaya & Team, 2020), and vehicle motion prediction (Caesar et al., 2020). In partieular, we will assess our method in long- term extrapolation tasks ranging from 60 to 200 timesteps. The detailed descriptions of these real-world datasets are presented in Appendix E.
5.1. Evaluation on Drone State Prediction
We first evaluate the performance of our method on the state prediction for a quadrotor drone, nonlinear system exhibiting complex oscillatory trajectories. Mean Absolute Error (MAE) and Mean Squared Error (MSE) metrics used to assess interpolation and extrapolation results.
As shown Table 1, our method achieves the best perfor- mance in both interpolation and extrapolation tasks. For interpolation, this improvement is attributed to our method's ability to capture sequence correlations by adjusting tra- jectories based on posterior estimation. For extrapolation, our Phy-SSM unit effectively learns generalizable physical dynamics from historical information using memory mecha- nisms, enabling precise predictions. Conversely, data- lata-driven continuous-time models perform relatively poorly on extrap- olation tasks, as they struggle to extract
For drone and COVID-19 modeling tasks, we compare our approach against state-of of-the-art (SOTA) methods in continuous-time modeling and physics- enhanced machine learning. Continuous- Time Models: 1) Latent ODE (RNN Encoder) (Chen et al., 2018), 8), 2) Latent ODE (ODE-RNN Encoder) (Rubanova et al., 2019), 3) Contiformer (Chen et al., 2024), 4) Simplified Structured Physics-Enhanced State Space Model (S5) (Smith et al.). Physics- Enhanced
Baseline Methods.
Machine Learning Methods: 1) GOKU (Linial et al., 2021), 2) Physics- Inteegrred VAE (Takeishi & Kalousis, 2021).
7

---

Physics-Enhanced State Space Model for Loon-TTerm Dynamics Forecasting
Table Performance comparison between our method and the baselines for In-Domain task using nuScenes dataset. All methods evaluated at 5% missing agent observations. The results are averaged over three random seeds. The best result is highlighted in bold black and second best highlighted in green.
Extrapolation (In domain) Task
Method
ADE
FDE
Speed Error |
Acceleration Error (x10*)
Error (x10²)
Wayformer
1.899:001114
4.954 0.086
31.396=3.766
57.080+
36.783+
AutoBot
2.474:10.954
6.322+1.658
2.291=00.985
2.432:1-06100
2.346e:1:0536
G2LTraj
2.425=0.553
4.847=1443
1.678=0.160
2.234:10.053
2.178=00081
GOKU
2.822==0.816
6.394=0.841
1.772=#0.539
2.439:10.019
18888+0.000
PIVAE
2.811=0.464
6.463=0.538
1.757=0.465
2,460
1.889=0.000
SDVAE
2.129100.055
5.706-10.20
1.589
2.390
1.904110.031
ODE2VAE
3.31810.269
6.98810.127
2.254.10.083
2.414110.024
1.88910.001
Ours
1.884-0.064
5.100+0.160
1.336+00003
2.399+H-0.022
1.884+0.001
Table Performance comparison of different methods for Out-of-Domanl task using nuScenes dataset. All methods are evaluated at 5% missing agent observations. The results are averaged over three random seeds. The best result is highlighted in bold black and the second hest is highlighted green.
Extrapolation (Out-of-domale Task
Method
ADE
FDB
Speed Error
Acceleration Error & (x10')
Jerk Error 4. (x10**
Wayformer
8.842.10.979
8.810 180
46.2331117.114
76.267.130.867
44.7291126144
141
AutoBot
11.366155.083
11.68314.335
3.780
3.366.12.484
2.716:1:.366
G2LTraj
10.755+22004
12.471+2.871
25.286+t-4.761
50.890=8.235
9.190=11401
GOKU
1.691+1.114
8.872=1.474
2.788+0.995
2.063++0.067
1.550++0.007
PIVAE
7.569+00.504
8.519=0.466
2.381++0254
2,081++.002
1.552++0.003
SDVAR
7.050+00.543
8.235-0.835
2.689+0.838
2.065-+0.034
1.549-+0.003
ODE2VAE
8.411:0.226
0,694=0.582
3.222410.233
2.075=10.022
1.550
Ours
7.197- 0.305
2.398+0.533
2.043++0.078
1.548
Ground Truth
SS
Ground Truth
GOKU
Ground Truth
Ours
1.5
1.0
12
1.0
Angular Velocity (radis)
Angulor Velocity (radis
Angular Velocity (radis)
1
0.0
0.0
-0.5
0
-0.5
-1.0
-1.5-
-15
-2.
Interp.
Vattr.
lajerp.
Extrn.
Interp.
Extra.
22.5
-2.0
400
1000
200
400
800
1000
200
400
600
800
1000
Time (ms)
Time (ms)
Time (ms)
(a) S5
(b) GOKU
(c) Ours
Figure 2. Trajectory plots of method and top two baseline models for drone angular velocity prediction along the x-axis. The performance is evaluated in both interpolation and extrapolation tasks, including (a) S5, (b) GOKU and (c) Ours. The left of the gray dashed line represents interpolation task (Interp.) while the right represents the extrapolation task (Extra.).
representations without inductive biases. While physics- enhanced baselines learn physics-consistent representations, their inability fully utilize subsequent observations hin- ders their capacity accurately model complex, oscillatory trajectories.
Additional full trajectory plots for all methods are provided in Appendix G.
5.2. Evaluation on COVID-19 Epidemiology Modeling
For the COVID-19 prediction task, we also use MAE and MSE as metrics assess the performance of our model. The results are presented in Table It can be observed
As shown in Figure 2, we visualize interpolation and ex- trapolation results for our method and the top two baselines.
8

---

Physics-Enhandee Enhanced State Space Model for Long-Term Dynamics Forecasting
Table Ablation studies using the drone dataset. Lower values indicate better performance. The results are averaged over three random seeds. The best result is highlighted in bold black and the second best is highlighted in green.
Interpolation Task
Extrapolation Task
Phy-SSM unit
Regularization
MAEJ (x10-L)
MSE (x10-3 2)
MAEI (x10) x10011
MSEJ (x10
x
1.059=00.22
3.091
8.426=1.351
17.333:44.84
X
0.927=0.057
1.860:10.155
3.008
2.176:10.079
Ours including both
1,002:+0004
2.228-+0.203
2.733-001 0.059
1.798++0.077
that our approach achieves the best performance in both interpolation and extrapolation tasks. The baselines do not perform well because they lack an effective mechanism to dynamically refine predicted trajectories for time-varying dynamical systems. In real-world epidemiology, some exter- nal factors such as temperature may influence the underlying dynamics of COVID-19 19 over time. In contrast, our method can dynamically refine predictions based on subsequent ob- servations and learn more accurate time-varying dynamics. Besides, we present the trajectory plots for our method and the baselines in Appendix G.
cussed above, our Phy-SSM demonstrates superior perfor- mance in long-term dynamics forecasting.
5.4. Ablation Studies
Effect of the Phy-SSM Unit. We first study the impact of the Phy-SSM Unit on prediction performance by removing it, transforming the model into a purely data-driven SSM. The results, shown in Table 5, demonstrate that excluding the Phy-SSM Unit significantly degrades extrapolation per- formance. In contrast, incorporating the Phy-SSM Unit significantly improves extrapolation performance compared to data-driven SSM alone.
5.3. Evaluation on Vehicle Motion Prediction
Effect of the Physics State Regularization Term. We also investigate the impact of physics state regularization term on model performance. In our experiment, we remove in the overall objective in Eq. (12). As shown in Table 5, without it, the resulting model tends overly learn the latent states from observed trajectories. As a result, the model seems to improve interpolation performance but degrade its extrapolation ability significantly. In contrast, our Phy-SSM can enhance the long-term extrapolation performance since the physics state regularization term can guide the model to learn generalized physical dynamics.
In motion prediction task, we follow standard settings and evaluate performance using Average Displacement Error (ADE), Final Displacement Error (FDE), Speed Accelera- tion Error, and Jerk Error (Feng et al., 2025; Xu et al., 2023). Here, ADE and FDE measure the accuracy of the predicted positions, while Speed, Acceleration and Jerk Error evalu- ate the physical plausibility of predictions. Detailed metric calculations can be found in Appendix D.2.3.
For fair comparison, all state-of-th of-the-art art (SOTA) methods predict single trajectory in this task. Moreover, we cat- egorize the predictions into two scenarios: 1) In-domain extrapolation predictions: Predictions spanning from 0 to 5 seconds, aligning with the temporal window observed dur- ing training. 2) Out-of-domain predictions: Predictions spanning from 5 to 6 seconds, exceeding the time range dur- ing training and thus evaluating the model's generalization ability to unseen temporal domains.
5.5. Sensitivity Analysis
We also study how the hyperparameters in the loss func- tion in Eq. (12) affect the performance of Phy-SSM in Appendix H.
6. Conclusion
The experimental results for these two scenarios are reported in Tables 3 and 4. Our method achieves slightly better per- formance than the baselines in in-domain tasks. Moreover, demonstrates significantly better performance in out-of- domain predictions, where SOTA data-driven methods per- form poorly. All PEML methods achieve better results than purely data-driven SOTA methods in extrapolation tasks, particularly in physics-related metrics (Speed, Acceleration, and Jerk Error), showcasing the effectiveness of physics- enhanced mechanisms. Among them, our method achieves the best results in ADE and FDE metrics, highlighting its ability to capture sequence correlations effectively.
We proposed a generalizable method, called Phy-SSM, that incorporates partially known physics into state space mod- els for long-term dynamics forecasting complex envi- ronments. Specifically, we developed novel Phy-SSM unit to improve the learning of more generalized physics representations from observations. Then, a physics state regularization is introduced to further enhance long-term prediction performance. Extensive experiments on three real-worke application demonstrated the superiority of the proposed method over baselines in long-term interpolation and extrapolation tasks.
In summary, based on the three real-world applications dis-
9

---

Physics-Enhancee State Space Model for Long- Term Dynamics Forecasting
Acknowledgments
Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
Research reported this paper was sponsored part by NSF CPS 2311086, NSF CIRC 716152, and Faculty Re- search Grant at William & Mary 141446. We would also like to thank Yizhuo Chen for helpful discussions dynamics VAE.
Chen, Y., Lu, L., Karniadakis, G. E., and Dal Negro, L. Physics-inforni neural networks for inverse problems in nano-optics and metamaterials. Optics express, 28(8):
11618-1 1168-1 2020.
Impact Statement
Chen, Y., Ren, K., Wang. Y., Fang. Y., Sun, W., and Li, D. Contiformer: Continuous-time transformer for irregular time series modeling. Advances in Neural Information Processing Systems, 36, 2024.
This paper aims to facilitate the application of machine learning in healtheare, physics, and cyber-physical systems while fostering interdisciplinary development. We believe this work holds significant potential to improve societal well- being, enhance the generalization ability of garning-enabbles cyber-physical systems, and promote collaboration across diverse fields. We foresee no negative societal implications arising from this research and consider its broader impact be highly positive.
Chung, J., Kastner, K., Dinh, L.., Goel, K., Courville, A. C., and Bengio, Y. A recurrent latent variable model for se- quential data. Advances in neural information processing systems, 28, 2015.
Cicirello, A. Physics-enhanced machine learning: posi- tion paper for dynamical systems investigations. arXiv preprint arXiv:2405.05987, 2024.
References
Anderson, R. M. Discussion: the kermack mekkndrrckk epidemic threshold theorem. Bulletin of mathematical biology, 53(1):1-32, 1991.
Cranmer, M., Greydanus, S., Hoyer, S., Battaglia, P., Spergel, D., and Ho, S. Lagrangian neural networks. arXiv preprint arXiv:2003. 04630, 2020.
Bacsa, K., Lai, Z., Liu, W., Todd, M., and Chatzi, E. Sym- plectic encoders for plysics-constrained variational dy- namics inference. Scientific Reports, 13(1):2643, 2023.
Dabrowski, J. J. and Rahman, A. Sequence-to-sequenee
imputation of missing sensor data. In Al 2019: Advances in Artificial Intelligence: 32nd Australasian Joint Con- ference, Adelaide, SA, Australia, December 2-5, 2019, Proceedings 32, pp. 265-276. Springer, 2019.
Baydin, A. G., Pearlmutter, B. A., Radul, A. A., and Siskind, J. M. Automatic differentiation in machine learning: a survey. Journal of machine learning research, 18(153): 1-43, 2018.
Eschmann, J., Albani, D., and Loianno, G. Data-driven sys- tem identification of quadrotors subject to motor delays. arXiv preprint urXiv:2404.07885 2024.
Bonfanti, A., Santana, R., Ellero, M., and Gholami, B. On the generalization of pinns outside the training domain and the hyperparameters influencing it. Neural Comput- ing and Applications, 36(36) 2024.
Faroughi, S. A., Pawar, N., Fernandes, C., Raissi, M., Das, S., Kalantari, N. K., and Mahjour, S. K. Physics-guided, physics-informed, and physics-encoded neural networks scientific computing, arXiv preprint arXiv:2211.07377 2022.
Brockman, G. Openai gym. arXiv: 1606:006000154000 2016.
arXiv preprint
Brunton, S. L., Proctor, J. L., and Kutz, J. N. Sparse iden- tification of nonlinear dynamics with control (sindye). IFAC-PapersOnLine, 49(18):710-715 2016.
Feng. L., Bahari, M., Amor, K. M. B., Zablocki, É., Cord, M., and Alahi, A. Unitraj: A unified framework for scalable vehicle trajectory prediction. arXiv preprint arXiv:2403. 15098, 2024.
Caesar, H., Bankiti, V., Lang, A. H., Vora, S., Liong, V. E., Xu, Q. Krishnan, A., Pan, Y., Baldan, G., and Beijbom, O. nuscenes: A multimodal dataset for autonomous driving. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp. 11621-11631, 2020.
Feng, L.., Bahari, M., Amor, K. M. B., Zablocki, É., Cord, M., and Alahi, A. Unitraj: A unified framework for scal- able vehicle trajectory prediction. In European Confer- ence Computer Vision, pp. 106-123, Springer, 2025.
Champion, K., Lusch, B., Kutz, J. N., and Brunton, S. L. Data-driven discovery of coordinates and governing equa- tions. Proceedings of the National Academy of Sciences, 116(45):224452222) 2019.
Gentine, P., Pritchard, M., Rasp, S., Reinaudi, G., and Ya- calis, G. Could machine learning break the convection parameterization deadlock? Geophysical Research Let- ters, 45(1 (5(11))5742-57)) 2018
10

---

Physics-Enhanced State Space Model for Long-Terae Term Dynamics Forecasting
Girgis, R., Golemo, F., Codevilla, F,, Weiss, M., D'uza, J. A., Kahou, S. E., Heide, F., and Pal, C. J. Latent variable sequential set transformers for joint multi-agent motion prediction. International Conference on Learning Representations (ICLR), 2022.
Kingma, D. P. Auto-encoding variational bayes. arXiv preprint arXiv: 14,2013, 2013.
Kong, J., Pfeiffer, M., Schildbach, G., and Borrelli, Kine- matic and dynamic vehicle models for autonomous driv- ing control design. In 2015 IEEE intelligent vehicles symposium (IV), pp. 1094-1099 IEEE, 2015.
Girin, L.. Leglaive, S., Bie, X., Diard, J., Hueber, T., and Alameda-Pineda. X. Dynamical variational au- toencoders: A comprehensive review. arXiv preprint (rXiv:2008.12595, 2020.
Li, Q. Peng, Z. M., Feng, L., Liu, Z., Duan, C., Mo, W., and Zhou, B. Scenarionet: Open-source source platform for large- scale traffic scenario simulation and modeling. Advances in neural information processing systems, 36, 2024.
Greydanus, S., Dzamba, M., and Yosinski, J. Hamiltonian neural networks. Advances in neural information process- ing systems, 32, 2019.
Linial, O., Ravid, N., Eytan, D., and Shalit, U. Generative ode modeling with known unknowns. In Proceedings of the Conference on Health, Inference, and Learning, pp. 79 04, 2021.
Gu, A. and Dao, T. Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint MrXiv:2312.00752 2023.
Lu, L., Pestouric, R., Yao, W., Wang, Z., Verdugo, F,, and Johnson, S. Physics-informet neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, B3(6):B1100-BI B1122, 2021.
Gu, A., Dao, T., Ermon, S., Rudra, A., and Ré, C. Hippo: Recurrent memory with optimal polynomial projections. Advances in neural information processing systems, 33: 1474-1487, 2020.
Lu, Y. and Lu, J. A universal approximation theorem of deep neural networks for expressing probability distributions. Advances in neural information processing systems, 33: 3094-3105, 2020.
Gu, A., Goel, K., and Ré, C. Efficiently modeling long sequences with structured state spaces. arXiv preprint arXiv:2111.00396 2021.
Gu, A., Goel, K., and Ré, C. Efficiently modeling long sequences with structured state spaces, 2022. URL aittps://arae
Lutter, M., Ritter, C., and Peters, J. Deep lagrangian net- works: Using physics as model prior for deep learning. In International Conference on Learning Representations, 2018.
org/ abs///l1.... 00396.
Huang, Z., Sun, Y., and Wang, W. Generalizing graph ode for learning complex system dynamics across envi- ronments. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 8088-09,,00 2023.
Mao, Y., Gu, Y., Sha, L., Shao, H., Wang, Q., and Abdelza- her. T. Phy-taylor: Partially deep neural networks via nn editing. IEEE Transactions on Neural Networks and Learning Systems, 2023.
lonides, E. L., Bretó, C., and King, A. A. Inference for non- linear dynamical systems. Proceedings of the National Academy of Sciences, 103(49):18438-11)
Nayakanti, N., AI-Rfou, R., Zhou, A., Goel, K., Refaat, K. S., and Sapp, B. Wayformer: Motion forecasting via simple & efficient attention networks. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 2980-2987. IEEE, 2023.
2006.
Kaufmann, E., Bauersfeld, L.,, Loquercio, A., Müller, M., Koltun, V., and Scaramuzza, D. Champion-level drone racing using deep reinforcement learning. Nature, 620 (79-6)9982-987, 2023.
Nicho, J. The sir epidemiology model in predicting herd immunity. Undergraduate Journal of Mathematical Mod- eling: One+ Two, 2(2):8, 2010.
Kidger, P., Morrill, J., Foster, J., and Lyons, T. Neural controlled differential equations for irregular time series. Advances in Neural Information Processing Systems, 33: 6696-6700 2020.
O'Driscoll, P., Lee, J., and Fu, B. Physics enhanced artificial intelligence. arXiv preprint arXiv:1903 1903.04442, 2019.
Kim, J., Lee, K., Lee, D., Jhin, S. Y., and Park, N. Dpm: A novel training method for physics-ifformed neural net- works in extrapolation. In Proceedings of the AAAI Con- ference on Artificial Intelligence, volume 35, pp. 8146- 8154, 2021.
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019.

---

Physics-Enhanced State Space Model for Long- Term- Dynamics Forecasting
Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physies- informed 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.
Yang, T.-Y,, Rosca, J., Narasimhan, K., and Ramadge, P. J. Learning physics constrained dynamics using autoen- coders. Advances in Neural Information Processing Sys- tems, 35:17157-17172 2022.
Yildiz, C., Heinonen, M., and Lahdesmaki, H. Ode2vae: Deep generative second order odes with bayesian neural networks. Advances in Neural Information Processing Systems, 32, 2019.
Raissi, M., Yazdani, A., and Karniadakis, G. E. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 2020.
Yin, Y., Le Guen, V., Dona, J., de Bézenac, E., Ayed, I., Thome, N., and Gallinari, P. Augmenting physical models with deep networks for complex dynamics forecasting. Journal of Statistical Mechanics: Theory and Experiment, (12))12)224012 2021.
Rajamani, R. Vehicle dynamics and control. Springer Sci- ence & Business Media, 2011.
Rubanova, Y., Chen, T., and Duvenaud, D. K. Latent ordinary differential equations for peagularly-samplee time series, Advances in neural information processing systems, 32, 2019.
Yu, J., Lu, L.., Meng, X., and Karniadakis, G. E. Gradient- enhanced thlysics-informes neural networks for forward and inverse pde problems. Computer Methods in Applied Mechanics and Engineering, 393:1 14823, 2022.
Smith, J. T., Warrington, A., and Linderman, S. Simplified state space layers for sequence modeling. The Eleventh International Conference Learning Representations.
Zhang, Z., Hua, Z., Chen, M., Lu, W., Lin, B.. Cai, D., and Wang, W. G2ltraj: A global-to-loca genera- tion approach for trajectory prediction. arXiv preprint arXiv:2404 19300093,,,...1.1111111.111.0011 2024.
Sohn, K., Lee, H., and Yan, X. Learning structured output representation using deep conditional generative models. Advances in neural information processing systems, 28, 2015.
Zhong, Y. D., Dey, B., and Chakraborty, A. Symplectic ode-net: Learning hamiltonian dynamics with control. In International Conference on Learning Representations.
Takaya, H. and Team, C. D. Covsirphy version [version number]: Python library for covid-19 analysis with phase- dependent sir-derived ode models, 2020. URL ht ttsps: //github. com/ wwwphillr/cooiiieeeeeeaeeeeeeeee 9-sir. Ac-
cessed: 2025-1-1.
Takeishi, N. and Kalousis, A. Physics-integrated variational autoencoders for robust and interpretable generative mod- eling. Advances in Neural Information Processing Sys- fems, 1801-42104,,,,,, 2021.
Tustin, A. A method of analysing the behaviour of linear systems in terms of time series. Journal of the Institution of Electrical Engineers- IIA: Automatic Regulators and Servo Mechanisms, 94(1):130-144 1947.
Wang, Y., Xiong, C., Wang. Y., Xu, P., Ju, C., Shi, J., Yang, G., and Chu, J. Temperature state prediction for lithium- ion batteries based on improved physics informed neural networks. Journal of Energy Storage, 73:10886; 2023.
Wehenkel, A., Behrmann, J., Hsu, H., Sapiro, G., Louppe, G., and Jacobsen, J.-H. Robust hybrid learning with expert augmentation. Transactions on Machine Learning Research, 2023.
Xu, D., Chen, Y., Ivanovic, B., and Pavone, M. Bits: Bi- level imitation for traffic simulation. 2023 IEEE Inter- national Conference on Robotics and Automation (ICRA), pp. 2929-2922222 IEFE, 2023.
12

---

Physics-Enin Eonsies-Enhaneed State Space Model for Long-Term Term Dynamics Forecasting
A. COVID-19 Visualization Example
In this section, we compare the performance three methods on COVID-19 dataset: purely data-driven deep state space models (SSMs) (Smith al.), the physics-enhanced neural ordinary differential equation (NODE) method (Linial et al., 2021), and our proposed method. The first 160 irregularly recorded days of infectious population data are fed into the model to predict results for the subsequent 0 to 240 time steps.
As shown in Fig. 3, the physics-enhanced NODE performs relatively well during the first 50 time steps. However, its performance declines significantly in the subsequent predictions due to its heavy dependence on initial conditions, lacking a mechanism to refine predictions using subsequent observations. The purely data-driven state space model can capture input sequence correlations and performs well interpolation tasks. However, it produces physics- irrational outputs in extrapolation tasks.
In contrast, proposed method effectively captures long-term input sequence correlations while integrating partially known physics knowledge to learn more generalized representations. This enables it to excel in long-term dynamical prediction tasks, even under noisy and irregular conditions.
le6
1,50
Ground Truth
Ground Truth
Ground Truth
Prediction Interpolation
Prediction Interpolation
Prediction Interpolation
1.25
Prediction Extrapolation
Prediction Extrapolation
Prediction Extrapolation
Infected Population
Interpolation Error
Interpolation Error
Interpolation Error
1.00
Extrapolation Error
Extrapolation Error
Extrapolation Error
0.75
0.50
0.25
0.00
0
50
100
150
200
250
50
100
150
200
250
0
50
100
150
200
250
Days
Days
Days
(a) Purely data-driven model
(b) Physics-enhanced NODE
(c) Our Phy-SSM
Figure 3. An illustrative example of predicting infectious COVID-19 population Spain (Takaya & Team, 2020). Data from 160 irregularly recorded days are used as inputs for models to predict to 240 days of infectious population. (a) Purely data-driven SSMs (Smith al.) capture sequence correlations effectively and perform well interpolation tasks but struggle with extrapolation tasks. (b) Physics-enhanced NODE (Linial et al., 2021)) can enhance prediction performance but still suffers from emror accumulation for extrapolation tasks. Our method captures long-tern sequence correlations and performs well in both interpolation and extrapolation tasks.
13

---

Physics-Enin Ennaace-Enhaneed State Space Model for Long-Term Dynamics Forecasting
B. Notations
In this section, we present the main notations used throughout the paper following table. Scalars are represented by lowercase letters (e.g., æ),, vectors by boldface lowercase letters (e.g., x), and matrices by boldface uppercase letters (e.g.. A, B).
Table 6. Summary of notations
Notation
Definition
71
number of data points
Noisy observations
Predicted observations
2
Latent system state
2
Extended latent system state
u
Control input
h
Memory hidden states
(z)
Extended system states
f
System dynamics
fku
Known system dynamics
f funk
Unknown system dynamics
Emission function
4
Sequential encoder
Decoder
d
Dimension of observations
d2
Dimension of system states
du
Dimension of control inputs
X' {x [0,7] + R
Observed trajectory set
u {u: [0,
Rd
Control input signal set Banach space Set of all system dynamics functions
F
f E F
A
System dynamics matrix
Á
System dynamics matrix in discretized form
knw
Known system dynamics matrix
A
Unknown system dynamics matrix
B
Control input matrix
B
Control input matrix in discretized form
unk
Unknown control input matrix Knowledge mask
M
O
Hadamard product
14

---

State Space Model for Long-Term Dynamics Forecasting
C. Proofs for Proposition
In this section, we theoretically analyze the uniqueness of the decomposition the dynamical system in Eq. (7) by solving the objective function min L in Eq. (12). Before that, we make the following assumptions:
Assumption 1. Based on the universal approximation theorem for probability distributions by neural networks (Lu & Lu, 2020), we assume that the encoder (o(+) and decoder Φ(-). parameterized by neural networks, are well-approximated or that the approximation error is at least bounded.
Assumption 2. There exists one and only one underlying dynamics f in Eq. (7) that minimizes the loss function C in Eq. (12)
Under these assumptions, we provide the detailed proofs for Proposition
Proofs of Proposition I. For the dynamical system in Eq. (7), we obtain the following equation (same as Eq.(8)) by extending the original state
dz(t) = A(t)=(t) + ((t)u(t) = (Arow + Auni (t)) + (tni(t)
dt
where 2 = === TT E Rd the extended state, and <b(=) E Rd denotes the additional extended terms. On the right hand side, Akow (t) captures known physical dynamics, whereas Annk (t) and Bunk (t) models unknown dynamics.
First, we try define the element wise form of the state matrix At)). By construction, no entry appears in both Axuw (t) and Aunk (4) simultaneously; in other words, they separate the nonzero entries of A().. Formally, for the i-th rows in Aknw (t) and (t) (i (1....,de }), they can be defined
(knw)
j E knw
So, (unk) ais
if J Juw
Akuw (t))ks
0,
j TH
[tunk()))as
=
if Jlu]
(16)
where Jhi and I disjoint index sets for the i-th row:
J = (1(1) 12 Jli) (l....,dd)) (
(1)
1
(9), (()
(1,..,,dsh, (b: = l...,, and m < dg),
unk-
As result, the i-th row of the state matrix A() can be expressed as
(kow)
if J knw
ais
[A()).
(17)
(unk)
if E J(i)
ais
Then, we try to demonstrate the uniqueness of the decomposition using proof of contradiction. More specifically, suppose that there exist another way to decompose A(t) into Akaw(t) and Aunk(t). Similar to Eq. (17), the i-th row of A() can be expressed as
(kuw) di (unk)
if J(6 knw
[A(t))
if E (() unke
(knw) * dij where a!Uy
(unk) * 0g are entries in Axnw and Aunk respectively. According to Assumption 2, there (knw) and dij
(unk)
only exists one state matrix At)) and input matrix Btt) = Bunk (t) that minimizes the loss function L in Eq. (12). Thus, for each element in the extended state ž, we can obtain
Cigj
(knw)
(1)
(kuw) (1)
Z +
Σ JEJun
(unk) (1) Σ = O, i i= 1, 1..,...,d (a (1)-a;
je
(kow) and (unk) * (unk) the above equation holds only when z(t) 0. This contradicts the fact that z represents the extended states, which cannot be zero all the time. Thus, the assumption that there exists two different ways to decompose A(t) yields a contradiction, proving the uniqueness of the solution.
Since (knw) # oi 02, di.i
15

---

Physics-Enhancee State Space Model for Long-Term Dynamics Forecasting
D. Detailed Experimental Settings
We present the detailed experimental settings this Section. All experiments are conducted on a server equipped with NVIDIA A6000 GPUs, utilizing the PyTorch framework (Paszke et al., 2019).
D.I. Hyperparameters for Models
This subsection provides details about the hyperparameters used for all the models.
To evaluate the performance of Phy-SSM on the drone state prediction, COVID-19 modeling task, and video pendulum prediction, we compared it with state-of-the-an continuous-time models and physics-enhanced machine learning methods. For the vehicle motion prediction task, in addition to physics-enhanced machine learning baselines, we included three additional state-of-the-art data-driven methods specifically designed for this task.
The baselines are listed as follows:
Continuous- Time Models: 1) Latent ODE (RNN Encoder) (Chen et al., 2018), 2) Latent ODE (ODE-RNN Encoder) (Rubanova et al., 2019), 3) Contiformer (Chen et al., 2024). 4) Simplified Structured State Space Model (S5) (Smith et al.).
Machine Learning Methods: 1) GOKU (Linial al., 2021), 2) Physics-Integrane VAE (Takeishi & Kalousis, 2021), 3) Symplectic DVAE (Bacsa et al., 2023). 4) ODE2VAE (Yildiz et al., 2019)
Data-driven Vehicle Motion Prediction Models:1) Wayformer (Nayakanti et al., 2023), 2) AutoBot (Girgis et al., 2022), and 3) G2LTraj (Zhang et al., 2024).
For fair comparisons, we controlled the number of parameters across all models to be equivalent. For all NODE-based baselines, to model the influence of time-varying control inputs on certain tasks, we concatenated the output of additional control encoder with the NODE solutions produce the final output, similar the operation in conditional VAE (Sohn et al., 2015). Furthermore, the Dopri5 method was selected as the ODE solver for all experiments. For Latent ODE (ODE-RNN Encoder), Contiformer, and data-driven vehicle motion prediction models, we used the hyperparameters from their original papers due to their specific architectural designs.
Below, we list the detailed hyperparameters used for the rest of the models in each experiment:
Drone State Prediction:
Latent ODE (RNN Encoder): The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by a 5-layer RNN with 32 hidden states. The decoder is a 4-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by a 3-layer MLP with 200 hidden units per layer. The control input encoder is a 2-layer MLP with 200 hidden units per layer.
$5: The encoder consists of 4-layer MLP with 200 hidden units per layer, followed by a 5-layer SSM with 128 hidden states. The decoder is a 4-layer MLP with 200 hidden units per layer. The control input encoder is a 2-layer MLP with 200 hidden units per layer.
The encoder consists of layer MI. with 200 hidden units per layer, followed by 5-layer RNN with 16 hidden states. The unknown parameters are obtained by 5-layer bidirectional LSTM with 32 hidden units per layer. The decoder a 4-yerr MLP with 200 hidden units per layer. The unknown dynamics are parameterized by a 3-layer MLP with 200 hidden units per layer. The control input encoder is 2-layer MLP with 200 hidden units per layer.
GOKU:
VAE: The encoder consists of a 4-layer MIP with 200 hidden units per layer, followed by a 5-layer RNN with 16 hidden states. The unknown parameters are obtained by a 5-layer bidirectional LSTMs with 32 hidden units per layer. The decoder is 4-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by 3-layer MLP with 200 hidden units per layer. The control imput encoder is 2-layer MLP with 200 hidden units per layer. The regularization loss hyperparameters follow the settings specified in the original paper.
Symplectic DVAE: The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by 5-layer RNN with 128 hidden states. The energy conservation dynamics are parameterized by a 5-layrr MLP with 128 hidden units per layer. The decoder is 4-layer MLP with 200 hidden units per layer. The control input encoder is a 2-layer MLP with 200 hidden units per layer.
16

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by 5-layer RNN with 128 hidden states. The latent ODE dimensionality is set to 10. The Bayesian neural network (BNN) comprises two layers with 50 hidden units each. The decoder is 4-layer MLP with 200 hidden units per layer. The control input encoder is 2-layer MLP with 200 hidden units per layer.
ODE2VAE:
Phy-SSM: The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by 5-layer SSM with 128 hidden states. The unknown dynamics are parameterized by a 4-layer SSM with 128 hidden units per layer. The decoder is a 4-layer MI.P with 200 hidden units per layer. The hyperparameters are set as 3 = 1 and X = 100.
COVID-19 Modeling:
Latent ODE (RNN Encoder): The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 32 hidden states. The decoder is 2-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by 3-layer MI. with 200 hidden units per layer.
S5: The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by a 4-layer SSM with 128 hidden states. The decoder is 2-layer MLP with 200 hidden units per layer.
The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 16 hidden states. The unknown parameters are obtained by 4-layer bidirectional LSTM with 32 hidden units per layer. The decoder is a 2-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by a 3-layer MLP with 200 hidden units per layer.
GOKU:
Physics- possics-Integrated VAE: The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 16 hidden states. The unknown parameters are obtained by a 4-layr bidirectional LSTM with 32 hidden units per layer. The decoder is 2-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by a 3-layer MI.P with 200 hidden units per layer. The regularization loss hyperparameters follow the settings specified in the original paper.
The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 128 hidden states. The energy conservation dynamics are parameterized by a 4-layer MLP with 128 hidden units per layer. The decoder is 2-layer MLP with 200 hidden units per layer.
Symplectic DVAE:
ODE2VAE:
The encoder consists of a 3-layer MLP with 200 hidden units per layer, followed by a 4-layer RNN with 128 hidden states. The latent ODE dimensionality is set 10. The BNN comprises two layers with 50 hidden units each. The decoder is a 2-layer MLP with 200 hidden units per layer.
Phy-SSM:
The encoder consists of 3-layer MLP with 200 hidden units per layer, followed by a 4-layer SSM with 128 hidden states. The unknown dynamics are parameterized by 3-layer SSM with 128 hidden units per layer. The decoder is 2-layer MLP with 200 hidden units per layer. The hyperparameters are set as / = 1 x 10-4 and A = x 10-4
Vehicle Motion Prediction:
The encoder consists of a 2-layer RNN with 256 hidden states. The unknown dynamics are parameterized by 4-layer MLP with 200 hidden units per layer. The decoder is 2-layer MLP with 256 hidden units per layer. The control embedding is extracted using off-the-shelf scene encoder (Nayakanti et al., 2023) and fused with latent states through cross-attention.
GOKU:
Physics-Integrated VAE: The encoder consists of a 2-layer RNN with 256 hidden states. The unknown dynamics are parameterized by 4-layer MLP with 200 hidden units per layer. The decoder is a 2-layer MLP with 256 hidden units per layer. The control embedding is extracted using off-the-shelf scene encoder (Nayakanti et al., 2023) and fused with latent states through cross-attention. The regularization loss hyperparameters follow the settings specified in the original paper.
Symplectic DVAE: are parameterized by a 2-layer MLP with 256 hidden units per layer. The decoder is a 2-layer MLP with 256 hidden units per layer. The control embedding is extracted using an off-the-shel scene encoder (Nayakanti et al., 2023) and fused with latent states through cross-attention.
The encoder consists of a 2-layer RNN with 256 hidden states. The energy conservation dynamics
17

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
The encoder consists of 2-layer RNN with 256 hidden states. The latent ODE dimensionality is to 10. The BNN comprises two layers with 50 hidden units each. The decoder is a 2-layer MLP with 256 hidden units per layer. The control embedding is extracted using an off-the-shelf scene encoder (Nayakanti et al., 2023) and fused with latent states through cross-attention.
ODE2VAE:
Phy-SSM: The encoder consists of 2-layer SSM with 256 hidden states. The unknown dynamics are parameterized by 4-layer SSM with 256 hidden units per layer. The decoder is 2-layer MLP with 256 hidden units per layer. The control embedding extracted using off-the-shelff scene encoder (Nayakanti et al., 2023) and fused with latent states through cross-attention. The hyperparameters are set as 3 = 1 and À = 1 x 10a.
Video Pendulum Prediction:
The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with hidden states. The decoder is 4-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by 3-layer MLP with 200 hidden units per layer. The control imput encoder is a 3-layer MLP with 200 hidden units per layer.
Latent ODE (RNN Encoder):
S5: The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by a 4-layer SSM with 128 hidden states. The decoder a 4-layer MLP with 200 hidden units per layer. The control input encoder a 3-layer MLP with 200 hidden units per layer.
The encoder consists of a 4-yerer MLP with 200 hidden units layer, followed by 4-layer RNN with 16 hidden states. The unknown parameters are obtained by a 4-layer bidirectional LSTM with 32 hidden units per layer. The decoder a 4-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by a 3-layer MLP with 200 hidden units per layer. The control imput encoder is a 3-layer MLP with 200 hidden units per layer.
GOKU:
Integrated VAE: The encoder consists of 4-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 16 hidden states. The unknown parameters are obtained by a 4-layer bidirectional LSTM with 32 hidden units per layer. The decoder is 4-layer MLP with 200 hidden units per layer. The unknown dynamics are parameterized by 3-layer MLP with 200 hidden units per layer. The control input encoder is 3-layer MLP with 200 hidden units per layer. The regularization loss hyperparameters follow the settings specified in the original paper.
The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by 4-layer RNN with 128 hidden states. The energy conservation dynamics are parameterized by a 4-layer MLP with 128 hidden units per layer. The decoder is 4-layer MLP with 200 hidden units per layer. The control input encoder is 3-layer MLP with 200 hidden units per layer.
Symplectic DVAE:
Phy-SSM:
The encoder consists of a 4-layer MLP with 200 hidden units per layer, followed by a 4-layer SSM with 128 hidden states. The unknown dynamics are parameterized by 3-layer SSM with 128 hidden units per layer. The unknown control influences are parameterized by 2-layer SSM with 128 hidden units per layer. The decoder is 4-layer MLP with 200 hidden units per layer. The hyperparameters are
10- and
18

---

Physics-Enhanced State Space Model for Mong-Term Dynamics Forecasting
D.2. Settings for Dynamical System
In this subsection, we provide details about the partially known physies equations for the quadrotor drone system, the SIR model for COVID-19, vehicle dynamics and video pendulum dynamics used in the experiments.
D.2.1. QUADROTOR DRONE SYSTEM
The real-world quadrotor drone dataset was collected by (Eschmann et al., 2024). The raw dataset includes three-axis angular velocity, angular acceleration, linear acceleration, and the four motor RPMs of the drone state. We preprocessed the data following the methodology outlined in (Eschmann et al., 2024), setting thrust (z-axis), geometric torque (x, y axes), and four motor RPMs as control-related inputs. All preprocessed data were normalized using z-score normalization. this task, our objective is to predict the three-axis angular velocity, angular acceleration, and linear acceleration of the drone.
We the standard dynamics equations (Kaufmann et al., 2023) for quadrotor, which are follows:
p v2
0
+
wb/2
1 R(q)
EraJe
0 m
+ g.
i=1
û R(q)is.
(18)
0b Once RR(q)
(19)
fi
Kig
wi Whng+
(20)
j=0 unknown
W
J wo) x wb
(21)
unknowwn
T
(Tp. x 1 trr Kr, f..
(22)
unknown
Wm
T
(W's wm)
(23)
unknown
Here, p and e represent the global position and velocity, respectively. and R(q) denote the orientation quaternion and the rotation matrix. fi represents the thrust produced by motor i, while wm and wap are the motor RPMs: state and setpoints, respectively. w; and Owc: denote the angular rate and body-frame acceleration, respectively. m and are the mass and gravitational acceleration. Tp., TJ and Ty, represent the position, force, and torque of the motor, respectively. J the inertia matrix, T., is the motor delay time constant, K,, is the torque coefficient of motor i, and Kj,, is the thrust coefficient (with exponent j) of motor i. The J, Tm. Kr, and K fiy are unknown in the system.
Following (Eschmann et al., 2024), we express the thrust curve using known variables, leading to Eq. (24):
4
R(q)
R(q)
Σ
Era.i + g.
m
4
8 m
+ R(q))-1.
(24)
i - R(g))
-1
19

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
Substituting Eq. (24) into Eqs. (18) and (19), derive the following body-frame acceleration equation.
acc
K 1 J.
(25)
Considering the practical assumption that body-frame acceleration depends on velocity and force across three axes, we perform state augmentation by adding a constant bias. Finally, the physics knowledge used in our work can be represented as:
dz
Az,
(26)
di
where:
1bb
W/
1]
0
P
0
m
0
0
0
A
0
0
0
0
0
0
0
0
00000
0
000
Wep
0
0
00000
*
Wap
0
Wap
0
Wap (*)
(+) denotes unknown parameters terms dependent the state and control inputs.
D.2.2. COVID-19 MODEL
The COVID-19 real-world dataset contains daily records of Confirmed (C), Infected (7), Fatal (F), and Recovered (Re) cases for each country. The number of Infected (Z) cases is derived as = = Re. Additionally, we utilized the 2020 population data (N) for each country from the Covsirphy library to compute the following metries:
Susceptible (S) Population (N) Confirmed (C)
Infected (I) Infected (I)
(27)
Removed (R) Fatal (F) + Recovered (Re)
Using the calculated S, I, and R variables, we first processed the data for each country by dividing all values by the total population N to ensure the value range is between 0 and 1. Subsequently, we applied z-score normalization to the processed input data.
To incorporate physics knowledge, we embed the SIR model (Anderson, 1991) into all models. The SIR model is a foundational compartmental model in epidemiology used simulate the spread infectious diseases. categorizes closed population into three compartments: Susceptible (S), Infectious (7), and Removed (R). The dynamics of these interacting groups governed by the following equations:
unknown
ds 8
ST
d/
N
unknown
d/
SI
N
unknown
dR
1,
dl
unknown
where S, J, and R denote the populations of the susceptible, infected, and removed groups (due recovery or death),
20

---

Physics-Enhanced State Space Model for Long-Term Term Dynamics Forecasting
respectively. The total population, represented by the constant N + / + R. remains unchanged. Here, 3 represents the contact rate between susceptible and infected individuals, while 7 denotes the removal rate of the infected population. Both parameters unknown, time-varying functions in real-world cases. As result, the following incomplete knowledge is used in the experiment.
S
(+)
0
S
d
dt
=
N
*
(28)
R
0
R
D.2.3, VEHICLE DYNAMICS
Based on Newton's 's second law of motion, the vehicle dynamics along the longitudinal and lateral axes are described as follows (Rajamani, 2011):
p m
F F,-
Facro
Aps - Rer
(29)
unknown
m i + v up)
Fus Fur
(30)
where p and y denote the vehicle's longitudinal and lateral positions, respectively, and tb is the yaw angle. The vehicle's mass is represented by m,, and Up is the longitudinal velocity. The forces F6 and F, are the longitudinal tire forces generated by the front and rear tires, respectively. The terms Rpf and Rpr represent the rolling resistance at the front and rear tires, while Facro accounts for aerodynamic drag along the longitudinal axis. Similarly, Fys and Fyr are the lateral tire forces exerted by the front and rear tires.
By defining the lateral velocity by ú and the yaw uib the system:
t/. derive the following state-space representation of
0
0
1
0
P
0
0
0
1
3/
0
0
0
0
0
//
0
+
0
+
8
(31)
dt
Up
0
8
0
Up
*
by
0
Bly
where the entries marked with (*) denote state- time-dependent terms that are unknown. The control inputs 0 and 8 represent throttle and steering, respectively.
Following (Mao et al., 2023), we adopt the practical assumption that throttle primarily depends on the longitudinal velocity and position, while the influence of steering remains less understood. Incorporating this prior knowledge, the state-space model is refined into the following form:
0
0
0
0
0
0
0
0
0
0
0 0
(32)
0
0
*
*
8
*
*
where the unknown terms (*) are assumed to be related to variations control inputs and state dynamics.
21

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
Evaluation Metric in Vehicle Motion Prediction To evaluate the performance of our proposed method, we utilize the following evaluation metrics in addition to MAE and MSE:
Average Displacement Error (ADE): The Average Displacement Error calculates the average Euclidean distance between the predicted trajectory and the ground truth trajectory over all time steps:
ADE
Σ pr Pr T
where I' is the trajectory length, P. is the predicted position at time t, and pa is the ground truth position at time t.
Final Displacement Error (FDE): The Final Displacement Error calculates the Euclidean distance between the final predicted position and the ground truth final position:
FDE
|p Prl12-
Speed Error: The L1 error of speed measures the average absolute difference between the predicted and ground truth speeds over all time steps:
Speed Error
2 0z Mels
T
where 0; and et represent the predicted and ground truth speeds at time respectively.
Acceleaation Error: The LI error of acceleration measures the average absolute difference between predicted and ground truth accelerations over all time steps:
Acceleration Error T
Σ ag arl
where à, and a, represent the predicted and ground truth accelerations time i, respectively.
Jerk Error: The LI error of jerk measures the average absolute difference between the predicted and ground truth jerks (rates of change of acceleration) over all time steps:
Jerk Error
2 1.: jel.
T
where and je represent the predicted and ground truth jerks at time respectively.
These metrics comprehensively evaluate the spatial accuracy of the predicted trajectories (ADE and FDE) and the dynamic properties of motion predictions (speed, acceleration, and jerk errors).
D.2.4. VIDEO PENDULUM DYNAMICS
In this toy experiment, we consider a general case where the pendulum system is affected by unknown friction and an unknown control input. Specifically, the system dynamics follow the equation:
dl(t) di
w()), dw(() sin O(t)
b
(33)
w(t)+ +
mlk A cos(2
df
unknown
unknown
where (t) represents the angular displacement of the pendulum, w()) denotes its angular velocity, is the gravitational acceleration, / is the length of the pendulum, is the damping coefficient, and m is the mass. and a denote the amplitude and frequency of the control input, respectively.
22

---

Physics-Enhancee State Space Model for LLon-TTerm Dynamics Forecasting
In this equation, the pendulum length I, the damping force caused by friction ((),, and the influence of the control input mi? all unknown. Next, we perform state augmentation by extending the nonlinear state terms such that s(t) sin (0(t)) and c(t) = cos (0(t)). This yields the following state-space model:
(t)
0
1
0
0
((t)
0
dt
(1)
s(t)
=
*
0
8
*
w(t)
(
s(t)
+
0
*
A cos (os(2)))))
(34)
c(t)]
0 (t)
c(t)
0
which serves as the physical knowledge used in this experiment.
For this dataset, we use the Pendulum-vO environment from OpenAl Gym (Brockman, 2016) to generate video data of the pendulum. We simulate 405 trajectories for training, 45 for validation, and 50 for testing. Each trajectory starts with a different initial condition and contains 300 time points with a time step of 0.05. The observed data are preprocessed such that each frame is resized 28 28 pixels and normalized to the range [0. 1] using min-max normalization.
To further increase the task's difficullyy, we add noise and introduce irregularly sampled settings. Specifically, zero-mean Gaussian noise with standard deviation of 0.3 is added to each pixel, and 20% of the data in each trajectory is randomly dropped. As result, each sequence contains 240 time steps, with the first 160 time steps used as model input for evaluating interpolation and the remaining 80 time steps used for evaluating extrapolation.
For the detailed parameter settings, we set m = 1.0, g = 10.0, and the damping coefficient 0.7. Additionally, for each trajectory, the pendulum length was uniformly sampled from [1, and the control input amplitude was uniformly sampled from [-5,5), 5), where negative values indicate control input in the opposite direction. These varying parameters, instead of being constant, make the task significantly more challenging.
D.3. Training Settings
For all experiments, the known dynamics Ako in our Phy-SSM unit is initialized using known physical parameters. The unknown dynamics Aunk is initialized based on the output of the deep SSM layer. For S5, adopt the default HiPPO initialization as described in (Gu et al., 2020).
Below, we provide the detailed training settings for each experiment:
Drone State Prediction: For all methods, we use the Adam optimizer with a learning rate 1 x 10 to train for a maximum of 20 epochs with a batch size of 64.
COVID-19 Modeling: For all methods, of 400 epochs with batch size of 32.
the Adam optimizer with learning rate of x 10-3 train for maximum
Vehicle Motion Prediction: For all baseline data-driven models, we strictly follow the training procedures and architectural configurations specified in the original works. For our method and the physics-enhanced machine learning models, we implement the AdamW optimizer with a cosine one-cycle learning rate schedule. Specifically, we set the maximum learning rate to 0.002 over 80 epochs with a batch size of 64. The scheduler parameters include a peak learning rate of 0.01, a percentage start of 0.01, a division factor of 10, and final division factor of 100.
Video Pendulum Prediction: For all methods, maximum of 150 epochs with batch size of 64.
the Adam optimizer with learning rate of 1 x 10-3 to train for
23

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
E. Real-World Datasets
We detail the three real-world datasets our experiments as follows.
(i) Drone state prediction: We use the real-world quadrotor drone dataset collected by (Eschmann al., 2024). The dataset includes three-axis angular velocity, angular acceleration, linear acceleration, and the four motor RPMs of the drone state. The data is irregularly and high-frequency recorded, nearly at 1010 Hz (minimum: 573.05 Hz. maximum: 1915.86 Hz). We split the data into 70%, 10%, and 20% for training, validation, and testing, respectively. The task is to use 800 timesteps of data to predict the states in the next 200 timesteps.
(ii) COVID-19 epidemiology modeling: We use the real-world COVID-19 19 dataset from Johns Hopkins University (JHU) provided by the Covsirphy Python library (Takaya & Team, 2020). The dataset contains daily records of the Susceptible, Infected, and Removed populations from various countries. For model training, we use the data from Armenia, Brazil, France, Germany, and Gabon. The United Kingdom dataset is used for validation while the data collected from Ireland and Spain used for testing. Notably, each country exhibits different unknown time-varying system dynamics, increasing the complexity of the task.
Additionally, we randomly dropped 10% of the recorded daily data to simulate missing records in real-world scenarios. A total 160 irregular data samples are used as the input of the model for predicting the next 80 irregularly required future days.
(iii) Vehicle motion prediction: For the vehicle motion prediction task, utilize nuScenes dataset (Caesar al., 2020) in the autonomous driving. This real-world dataset provides 2 seconds of past trajectories and seconds of future trajectories, with 5% missing agent observations. It includes detailed annotations such as velocity, heading, and position in 2D coordinate system. Additionally, this dataset offers high-definition (HD) maps containing lane boundaries, road centers, and traffic signals. We use an off-the-shelf scene encoder (Nayakanti et al., 2023) to extract environmental context as the control input for our model.
Before training, we preprocess and standardize the data 10 Hz using ScenarioNet (Li et al., 2024), following the approach in (Feng et al., 2024). The dataset is split into 80% for training and 20% for validation, while the nuScenes test file is used to evaluate the model performance. During training, the model takes the first 2 seconds of data as input to predict the subsequent 5 seconds. During testing, the prediction horizon is extended to seconds to further evaluate the model generalization.
24

---

Physics-Enian Plysics-Enhanced State Space Model for Long-Term Dynamics Forecasting
F. Additional Experimental Results Video Pendulum
addition, we present the evaluation results of different methods using video pendulum data. MAE and MSE are used as metrics to measure the performance of each method by comparing predicted framee with ground truth frames. The first 160 irregularly sampled frames are provided as input to the models to predict frames 0 to 240.
The results are shown in Table 7. For interpolation, our method effectively captures sequence correlations by adjusting trajectories based on subsequent observations, achieving competitive results. For extrapolation, our Phy-SSM unit learns generalizable physical dynamics and shows the best extrapolation performance.
In contrast, data-driven continuous- time models such as S5 and Contiformer perform well in interpolation tasks due to their ability to capture sequence correlations. However, they perform poorly in extrapolation tasks because they struggle extract physics-consisterd representations without inductive biases. Physics-enhanced baselines, on the other hand, learn physics- conssisc-nnn representations but fail to fully utilize subsequent observations. This limits their ability to learn generalized physical dynamics, resulting in poor long-term predictions under noisy and irregular data conditions.
Table 7. Performance comparison of different methods in terms of interpolation and extrapolation using pendulum dataset. The results are averaged over three random seeds. The lower better. The best result highlighted bold black and second best is highlighted in green.
Interpolation Task
Extrapolation Task
Method
MAE J. (x10-1)
MSE 4 (x10
MAE J. (x10-1)
MSE J (x10-²)
Latent ODE (RNN Enc.)
1.395=:0.018
2.713:10.053
1.188=00033
1.627=00.099
Latent ODE (ODE-RN) Enc.)
1.410=0.044
2.745::0.123
1.085=0.034
1.622:10.081
ContiFormer
1.136=00.022
1.350=:0.054
1.632=00001
5.165-+0.146
S5
1.156:10.018
1,44410,078
1.663=0.134
5.476=11.16
GOKU
1.38410.002.
2.694110.134
1.229=0.017
1.740:10.056
PI-VAE
1.399
2.8044100334
1.262=00019
1.826:00.036
SDVAE
1.338=00003
2.755=10.064
1.199
1.679=00.087
Ours
1.142:00002
1.409:0.0222
1.145=0.007
1.17:10.052
25

---

Physics-Enhanced State Space Model for Long-TTrr Dynamics Forecasting
G. Trajectory Plots for Different Methods
this section, we provide detailed trajectory plots for methods across each experiment. In the following subsections, we provide detailed trajectory plots for each method in each experiment.
G.I. Visualization Results of Drone State Prediction
The trajectory plots for methods on the drone state prediction task are provided in Fig. 4.
Ground Truth LatemoOEE (RNN Eac.)
Ground Trush
Ground Truth
LatenODD Enc.)
Contiformer
1.5
1.5
2
1.0
1.0
Angaler Velocity (radis)
Angular Volocity (radis)
Angular Velocity (rudis
0.5
0.5
0.0
0.0
110
1.5
2..
Interp.
Extra.
-2.5
Injerp.
Extra.
-2.0
Interp
Extra.
200
400
600
800
1000
200
400
600
800
1000
200
400
600
800
1000
Time (ms)
Time
(mu))
Enc.)
LatentODE (ODE-RNN Enc.)
(c) Contiformer
Ground Truth
SS
Ground Trath -.-.. GOKU
Ground Truth
PIVAE
1.5
1.5
1.0
Angular Veloeity (radis)
Angular Velucity (codis)
Angular Balaar Volocity (tadi))
0.5
0.0
0.0
11.0
-15
1.5
-2.0
Interp.
Vattra
Idjerp.
Extra.
2.0
Interp.
Extm.
-2.5
0
200
400
600
800
1000
-2
200
400
600
800
1000
200
400
600
800
1000
Time (ms)
Time (ms)
(ms)
(d) $5
(o) GOKU
(f) PIVAE
Ground Truth
SDVAB
Ground Truth
Ours
1.5
1.0
Angular Velocity (rabis)
0.5
Angular Velocity
0.0g
-0.5
-1.0
-1.5
Interp.
-2.0
1000
Time (mns)
Time (ms)
(g) SDVAE
(b) Ours
Figure 4. Trajectory plots our method and baseline models for drone angular velocity state prediction along the x-axis. The performance evaluated in both interpolation and extrapolation tasks, including (a) LatentDE (RNN Enc.), (b) LatentODE (ODE-RNN Enc.), (c) Contiformer, (d) S5, (e) GOKU, (f) PIVAF, (g) SDVAF, and (h) Ours. The left the gray dashed represents interpolation task (Interp.) while the right represents the extrapolation task (Extra.).
26

---

Mhysics-Enhancee State Space Model for Long-Term Dynamics Forecasting
G.2. Visualization Results of COVID-19
The trajectory plots for all methods on the COVID-19 epidemiology modeling task are provided in Fig. 5,
Ground Truth:
LatentODB E(c.)
LakerODDE (ODE-RNN
Contifornee
107
.77
1:7
4.725
4.725
Susupibble Population
4.700
Succppible Population
4,700
Susepsille Population
4. 4.700
4.675
4.675
4.675
4.650
4,650
4.650
A.625
4.625
4,500
4.600
4.600
4.575
Interp.
Extra
Interp.
Extra.
Interp.
Extra.
4,550
4,550
4.550
50
100
150 Days
200
250
50
100
150 Days
200
250
100
150
200
250
(a) Laten:ODE (RNN Ein.)
(b) Laten:DDE (ODE-RNN Enc.)
(c) Contiformer
Graund Trubh
GOKU
Ground Truth
4.725
4.725
Suscentble Population
4.700
Suscmtible Population
4,700
Suscppible Popaletion
4.700
8.675
4.675
8.675
4.650
6.650
4.650
4.625
4.625
4.625
4.600
4.600
4.600
4.575
4.575
4.575
Interp.
Extra.
Interp.
50
100
130
200
250
4,550
50
100
150
200
250
50
100
150 Days
200
250
Days
Days
(d) S5
(e) GOKL
( PIVAE
Ground Truth
SDVAE
Ground Truth
Our
167
4.750
4.725
4.725
4.700
Suscettile Populatio
Suscopable Popalation
4.700
4.675
4.675
4,650
4.650
4.625
4.625
4.600
4.600
4.575
Interp.
Extra.
Interp.
Extra.
4,550
4.550
50
100
150
200
250
0
50
100
150 Days
200
250
Days
(g) SDVAE
(h) Ours
Figure 5. Trajectory plots of our method and all baseline models for 19 susceptible population prediction in Spain. The performance evaluated in both interpolation and extrapolation tasks, including (a) LatentODE (RNN Enc.), (b) LatentODE (ODE-RNN Enc.), (c) Contiformer, (d) S5, (e) GOKU, (f) PIVAE, (g) SDVAE, and (h) Ours. The left of the gray dashed line represents interpolation task (Interp.) while the right represents the extrapolation task (Extra.).
27

---

Physicc-Enhanced State Space Model for Long-Term Dynamics Forecasting
G.3. Visualization Results of Vehicle Motion Prediction
The trajectory plots for all methods on the vehicle motion prediction task are provided in Fig.
Ground Truth
Wayformer
Ground Truth
Autobot
Coordinnate (m)
Coordinate
40.
36.
30
20
InND Ext.
Oun-D
In.D
DeDDD
Ext.
Ext.
Ext.
Y Coardinate (m)
2
Y Coordimate
1
In-D
Out-D
In-D
Out-D
Ext.
Fxt.
Ext.
Fxt.
Time (8)
Time (8)
(4) Wayformer
(b) Autobot
Ground Truth
G2LTraj
Ground Trath
GOKU
50
Cooninant (te)
30
Coordinste
40
40
30
30
20
In-D
Out-D Ext.
20
In-D
Out-D
10
Ext.
10
Ext.
Ext.
Y Coondinate (m)
Y Coordnatcc
Out-D Ext.
OOiED
2
Ext.
Exi.
Ext.
Time (s)
(c) G2L.Tai
(d) GOKU
Ground Truth
PIVAE
Ground Truih
Ours
50 40
50
X Coardinate (m)
50
X Coordinato
Chordinate
40
30
30
30
20
10
Ext.
Out-D
Ext.
20
10
In-D
Ext.
Out-D
Ext.
In-D Ext
Out-D
Ext
13 34
Y Conrdirate (m)
Y Conrdinste (m)
Conrinatto
In-D
Out-D
In-D
Out-D
In-D
Out-D
Ext.
2
Time (5)
Time (6)
Time (9)
(e) PIVAE
(0 SDVAE
(g) Ours
Figure Trajectory plots of our method and all baseline models for vehicle motion prediction. The performance evaluated both in-domain and out-of-domain extrapolation tasks, including (a) Wayformer, (b) Autobot, (c) G2ITraj, (d) GOKU, (e) PIVAE, (f) SDVAE, and (g) Ours. The left of the gray dashed line represents in-domain extrapolation task (In-D Ext.) while the right represents the out-of-domain extrapolation task (Out-D Ext.).
28

---

Physics-Enhann Ennaace- State Space Model for Loon-TTerm Dynamics Forecasting
H. Sensitivity Analysis
Lastly, we study how the hyperparameters in the loss function Eq. 12 affect the performance of Phy-SSM using the drone dataset. Our method involves two hyperparameters, 3 and A. As shown in Table 8, the results demonstrate that our method achieves consistently good performance when these hyperparameters are within an appropriate range, indicating that insensitive to variations in hyperparameter settings.
Table 8. Sensitivity analysis using drone dataset. All experiments were conducted using fixed random seed to ensure consistency.
Interpolation Task
Extrapolation Task
Hyperparameter
MAE (x10-3)
MSE 4 (x10-2)
MAE + (x10-1)
MSE (x10-
A = 0.1, À = 1
8,804
1.696
2.715
1.832
B 0.1, À 10
9.309
1.878
2.787
1.872
B 0.1, À 100
10.226
2.371
2.770
1.862
β İ 1
9.091
1.825
2.757
1.888
β Y 10
8.976
1.771
2.785
1.887
B = 1, = 100
9.641
2.024
2.692
1.745
3 = 10, A =
9.215
1.846
2.841
1.933
B = 10, À = 10
9.538
1.915
2.854
1.977
B 10, À 100
10.540
2.477
2.822
1.938
I. Guideline for Knowledge Mask Design
The knowledge mask is designed to distinguish which components of the system dynamics should be learned (unknown) and which are predefined (known). This allows the model to focus its learning capacity unknown physical terms while preserving known physical laws as hard constraints. Formally, the knowledge mask is a binary matrix M E {0, 1 1] d]dx applied via Hadamard product to the learned dynamics components. The refined unknown dynamics are computed as:
Aunk (1) MA C Aunie())
(35)
B Bunk (f) MB ④ Buuk((),
where Amnk (t) and Bunk (t) the raw outputs from the unknown dynamics learner. We categorize the dynamics terms in general system into three cases, and describe how the knowledge mask should be applied in each:
Fully known terms: These terms are derived from physical laws with known parameters (e.g., gravity). Their corresponding mask entries are set 0, preventing the model from updating them during training.
Fully unknown terms: These dynamics are not governed by any known physical law. Their corresponding mask entries are set to 1, allowing them be freely learned by the model.
Partially known (overlapping) terms: These contain both known and unknown components. In this case, the entire term is treated as "unknown" during learning (i.c., mask entry set to 1), and the known part is reintroduced in post-processing.
For example, in the COVID-19 model in Eq. (28), the first term can be expressed as - (*), where is known and (+) is unknown. We model the unknown component using the deep SSM, and multiply it by the known factor afterward to obtain the final expression.
29

---

Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting
J. Regularization Metric Experiments
conduct experiments comparing different distance metrics for the regularization penalty, including Chebyshev distance, cosine distance, and Euclidean distance. The results, presented in Table 9, show that the Euclidean distance achieves the best performance in extrapolation tasks. Chebyshev distance emphasizes worst-case deviations, while cosine distance captures directional similarity, which may not fully penalize magnitude differences. Since our objective is to measure the overall discrepancy between two physical state trajectories, Euclidean distance is not only empirically effective but also conceptually the most appropriate choice.
Table Performance comparison of different metrics used regularization term using drone dataset. The results are averaged over three random seeds. The lower the better. The best result highlighted in bold black and the second best is highlighted in green. Our method, which adopts Enclidean distance, achieves the best performance in extrapolation tasks.
Interpolation Task
Extrapolation Task
Method
MAE (x10-*)
MSE (x10-1)
MAE 4 (x10-1)
MSE (x10-1)
Chebyshev distance
3.957-10072
3.361110.19
4.342.10.050
4.440110.223
Cosine Distance
0.997=00.299
0.208:100012
3.01910.100
2.152:10.144
Euclidean distance
1.002:10.044
0.222:10.020
2.73310.059
1.798:10.079
30


Reviews:
[{"review_id": "NSUjYmgygD", "reviewer": "Reviewer_VtD1", "summary": "This paer proposes Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting in complex environments. Our motivation is that SSMs can effectively capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge improves generalization ability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "## Response to Reviewer VtD1\n**Q1.1**: About understanding the method’s complexity and presentation clarity. \n\n**A1.1**: To help you better understand our work, we outline the problem, motivation, and key contributions below.\n\nThis work addresses the problem of long-term dynamical forecasting in complex environments where data are noisy and irregularly sampled. Our motivation is that SSMs can effectively capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge improves generalization ability. \n\nOur key contributions can be summarized as follows:\n1) We propose Phy-SSM, a novel approach that integrates partially known physics into state-space models to improve generalization for long-term forecasting in complex environments. \n2) To enhance long-term prediction accuracy, we introduce a physics state regularization term that constrains latent states to align with system dynamics. \n3) We provide a theoretical analysis demonstrating the uniqueness of solutions in our framework.\n4) Extensive experiments on three real-world applications, including ve\nhicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting, demonstrate the superior performance of Phy-SSM over\nthe baselines in both long-term interpolation and extrapolation tasks.\n\nAdditionally, based on feedback from the other reviewers, we have:\n1) Included ODE2VAE as a new baseline to enable a more comprehensive comparison with uncertainty-aware NODE-based models, as shown in **Tables 1–4** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)].\n2) Added detailed ablation studies evaluating our choice of the L2 norm for regularization against alternative distance metrics (e.g., cosine distance, Chebyshev distance), as shown in **Table 5** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)].", "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": "I'm not an expert in this area, I want to know how complex it is and need a detailed presentation.", "other_comments_or_suggestions": ".", "questions_for_authors": ".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "."}}, {"review_id": "IYCmkDkd13", "reviewer": "Reviewer_ooWi", "summary": "This paper proposes a general-purpose framework that integrates partial physics knowledge into state space models. The topic is attractive and key innovation is clear.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "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": "## Response to Reviewer ooWi\n**Q2.1**: About validating physical integration beyond the ablation drone experiment, especially for nonlinear and multi-variable systems.\n\n**A2.1**: We evaluate our method on three real-world nonlinear, multi-variable systems: COVID-19, drone, and vehicle dynamics as presented in Tables 1–4 of our submission. Compared to data-driven models like S5, our physics-integrated approach consistently outperforms baselines, validating its effectiveness under diverse dynamical systems.\n\n**Q2.2**: About the disjoint assumption in Proposition 1 and the generalizability of the knowledge mask definition to overlapping dynamics.\n\n**A2.2**: The uniqueness result in Proposition 1 still holds when a term includes both known and unknown parts (e.g., a known multiplier applied to an unknown function). This is because we can treat the entire term as \"unknown\". For example, in the COVID-19 model (Eq. 28), the term $-\\frac{I}{N} \\cdot (*)$ is handled by learning the unknown part with a deep SSM and reintroducing the known factor $-\\frac{I}{N}$ in post-processing.\n\nThe same principle applies to the knowledge mask design. For disjoint dynamics, we assign 1 to unknown terms and 0 to known ones. For overlapping terms, the mask marks the full expression as unknown, with known components injected after training. We will introduce this guideline of mask in our revision.\n\n**Q2.3**: About using L2 regularization and other principled alternatives like Lagrangian multipliers.\n\n**A2.3**: We have done additional experiments using other metrics, as shown in **Table 5** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)]. We can see that L2 norm performs best. While Lagrangian multipliers are theoretically sound, they are difficult to apply in DNN models due to non-convexity and large searching space. We leave this for future work.\n\n**Q2.4**: About recent physics-guided transformers (e.g., PDE-Refiner) or Neural ODEs with uncertainty baselines, and the rationale for choosing SSMs.\n\n**A2.4**: We have included Contiformer [1], a state-of-the-art physics-based transformer, as a baseline. PDE-Refiner [2] is not suitable for irregularly sampled dynamics due to its reliance on fixed-step one-step MSE loss. Per your suggestion, we also added ODE2VAE [3], a Neural ODE with uncertainty-aware priors. Our method still achieves the best performance (see results in this [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)] **Tables 1–4**).\n\nWe choose SSMs as the core module of our approach because (i) they capture long-term dependencies via HiPPO memory, and (ii) their continuous-time formulation naturally handles irregular data.\n>[1] Chen et al. Contiformer. NeurIPS 2023\n>\n>[2] Lippe et al. Pde-refiner. NeurIPS 2023\n>\n>[3] Yildiz et al. Ode2vae. NeurIPS 2019\n\n**Q2.5**: About generalization of COVID-19 data and lack of large-scale benchmarks.\n\n**A2.5**: In the COVID-19 experiment, we train the model on data from six countries and evaluate on two unseen countries (Ireland and Spain), demonstrating generalization across regions.\n\nBeyond that, we have already included the large-scale autonomous driving (AD) dataset nuScenes [4], which contains 1000 driving scenes from Boston and Singapore—two cities known for dense traffic and challenging driving conditions. This dataset is widely used in AD research [5–7]. Experiment shows the generality of our method in learning vehicle dynamics from real-world data.\n>[4] Caesar et al. Nuscenes. CVPR 2020\n>\n>[5] Girgis et al. Autobot. ICLR 2022\n>\n>[6] Ren et al. Safety-aware motion prediction with unseen vehicles for autonomous driving. ICCV 2021\n>\n>[7] Zhang et al. G2LTraj. IJCAI 2024\n\n**Q2.6**: About the simplification of vehicle dynamics and the lack of actuator delay modeling.\n\n**A2.6**: Following prior works [8,9], we incorporate partially known vehicle dynamics into our Phy-SSM model. Experimental results show that our method performs the best. Nevertheless, we agree with the reviewer that accounting for actuator delay is a valuable direction, and we plan to extend our method to address this scenario in future work.\n>[8] Rajamani, Vehicle dynamics and control. 2011\n>\n>[9] Mao et al. Phy-Taylor. TNNLS 2023\n>\n\n**Q2.7**: About the formality of the uniqueness proof and the lack of broader theoretical discussions (e.g., stability).\n\n**A2.7**: We use a contradiction-based argument for proving uniqueness, which is a standard formal method. We will refine the proof in the revised version for clarity and rigor. \n\nAs for stability analysis, it is beyond our current scope, our focus is on learning the dynamics model rather than control design. we recognize its importance and leave it in future work.", "claims_and_evidence": "1. Phy-SSM effectively integrates partial physics into SSMs for improved generalization. The dynamics decomposition (Eq. 8) and knowledge masking (Eq. 9) are designed to enforce physical constraints. The idea is sound but the experiment is weak. Only the ablation results on drone state is not enough to state that it can integrate physical knowledge. How about other complex systems, especially with non-linear dynamics and multi-variable coupling? \n\n2. The physics state regularization term enhances long-term prediction accuracy. It seems to make sense.\n\n3. Theoretical guarantees ensure uniqueness of the dynamics decomposition. Proposition 1 assumes that the known and unknown terms occupy disjoint positions in the matrix (Equation 16). However, in practical systems, partial overlap may exist (e.g., certain terms containing both known and unknown components simultaneously). In such cases, uniqueness cannot be guaranteed, but the paper does not discuss this scenario.", "methods_and_evaluation_criteria": "Strengths:\n1. The Phy-SSM unit and knowledge masking mechanism provide a systematic way to embed physics into SSMs.\n2. Diverse applications (drone, COVID-19, vehicle) and metrics (MAE, MSE, ADE, physics-based errors) strengthen validity.\n\nWeaknesses:\n1. The process of defining knowledge masks (e.g., Eq. 15) is described for the pendulum example but lacks generalizable guidelines for other systems.\n2. The physics state regularization term (Eq. 12) is implemented as a simple L2 penalty; more principled approaches (e.g., Lagrangian multipliers for hard constraints) are unexplored.", "theoretical_claims": "Proposition 1 asserts uniqueness of the dynamics decomposition but relies on assumptions (e.g., disjoint support of known/unknown terms) without discussing their practical validity.", "experimental_designs_or_analyses": "1. Omission of recent physics-guided transformers (e.g., PDE-Refiner) or hybrid models limits comparative rigor. Why SSM is your choice?\n\n2. Small-scale datasets (e.g., COVID-19 data from only 8 countries) raise concerns about generalizability.\n\n3. The influence of control inputs (e.g., vehicle steering/throttle) is oversimplified; real-world actuator dynamics (e.g., delays) are unaddressed.", "relation_to_broader_scientific_literature": "The work builds on physics-informed ML (e.g., PINNs, Hamiltonian NODEs) and SSMs (e.g., S4, S5), advancing hybrid modeling for irregular dynamics.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "1. The uniqueness proof is informal, and broader theoretical implications (e.g., stability) are unexamined.\n2. Key competitors like Neural ODEs with uncertainty-aware priors or PDE-Refiner are absent.\n3. The knowledge masking heuristic may not scale to systems with overlapping known/unknown dynamics.\n4. Experiments lack large-scale benchmarks (e.g., climate modeling or robotics datasets) to test generality.", "other_comments_or_suggestions": "See review above", "questions_for_authors": "See review above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZVuumGaCfO", "reviewer": "Reviewer_xWcK", "summary": "The paper aims to improve long-term forecasting using state space models based on deep learning to a) embed prior knowledge about physical systems and b) handle noisy irregularly sampled data. Specifically, the paper proposes to separate the state matrix into known and unknown / learnable elements. To this end, the paper proposes to use a “binary knowledge mask” that effectively freezes elements in the state matrix that are known a priori. Similar to the literature in this area, to handle irregularly sampled data, the continuous-time state matrix is optimised, i.e., the system is discretised during each forward pass using Tustin discretisation.  A Variational Autoencoder is used for optimisation. In addition to the reconstruction loss and Kullback-Leibler divergence, the authors propose to introduce an addition regularisation term that encourages the encoder to adhere to the system dynamics. The proposed approach is evaluated for interpolation and extrapolation, and for three datasets that involve drone state prediction, COVID-19 epidemiology forecasting, and vehicle motion prediction. The results are compared against state-of-the approaches, including recurrent encoders, the contiformer, VAEs, and the S5 SSM. The results indicate that the proposed approach outperforms baseline approaches, particularly for the extrapolation tasks.  For vehicle motion prediction, the results are further divided into in-domain prediction (predictions within 0-5 seconds) and out-of-domain prediction (predictions from 5-6 seconds). Results indicate that the proposed approach performs particularly well for out-of-domain prediction compared to the baseline approaches.\n\n## Update after rebuttal:\nI raised my recommended score from 3 to 4.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "## Response to Reviewer xWcK\n**Q3.1**: About handling uncertainty in domain knowledge within our model.\n\n**A3.1**: We do not explicitly handle uncertainty in domain knowledge in this work. However, such uncertainty can be modeled within the unknown dynamics. A possible approach is to apply conformal prediction [1] as a post-processing step to estimate uncertainty in the deep SSM outputs. We will explore this in future work.\n\n>[1] Kamile et al. Conformal time-series forecasting. NIPS 2021\n\n**Q3.2** About the lack of ablations comparing L2 with other regularization metrics.\n\n**A3.2:** Following your suggestion, we have added comparisons with alternative metrics, including Chebyshev distance, and cosine distance. The results are presented in **Table 5** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)], where our method achieves the best performance in extrapolation tasks. We will include these results in the revision.\n\nChebyshev distance focuses on worst-case deviations, while cosine distance captures directional similarity, which may not fully penalize deviations in magnitude. Since our goal is to measure the overall discrepancy between two physical state trajectories, the L2 norm is not only empirically effective but also conceptually the most appropriate metric.\n\n**Q3.3**: About why MLP outperforms standard SSMs in ablation study, explain performance gains in extrapolation, and the placement of ablation study in the paper.\n\n**A3.3**: We would like to clarify that we do not replace the entire Phy-SSM unit with MLPs in the ablation study. Instead, we only replace the SSM layer of approximating unknown terms inside the Phy-SSM unit (highlighted in the blue rectangle in Fig. 1) with an MLP. we will include a clearer and more detailed explanation in the revised version.\n\nThe observed improvements—particularly in extrapolation—are due to the SSM layer’s ability to model long-range dependencies, which MLPs cannot effectively capture. This allows the Phy-SSM to learn more generalizable unknown physical dynamics, especially over extended horizons.\n\nFinally, per your suggestion, we will move the ablation study from the appendix into the main body of the paper.\n\n\n**Q3.4**: About typos and structural suggestions.\n\n**A3.4**: We will correct all listed typos and adjust the section order as suggested in the revised version.\n\n**Q3.5**: About the computational cost of Phy-SSM compared to standard SSMs and MLPs (Appendix H).\n\n**A3.5**: As clarified in A3.3, we do not replace the entire Phy-SSM unit with MLPs. To address your concern, we report the computational costs of the standard SSM, our Phy-SSM module, and a NODE-based baseline (GOKU). The results are provided in **Table 6** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)].\n\nPhy-SSM is slightly slower than the standard SSM due to the additional structure and regularization components introduced for physics integration. However, it is faster than NODE-based models, as it preserves the parallel-scan acceleration capability inherent in SSMs.\n\n**Q3.6**: About the initialization for the Phy-SSM module and the S5.\n\n**A3.6**: In the Phy-SSM module, the known dynamics $A_{knw}$ is initialized using known physical parameters. The unknown dynamics $A_{unk}$ is initialized based on the output of the deep SSM layer. For S5, we follow the default HiPPO initialization [2].\n\n>[2] Gu et al. \"Hippo: Recurrent memory with optimal polynomial projections.\" NIPS 2020", "claims_and_evidence": "The paper makes two main claims: 1) Embedding prior knowledge about the system dynamics improves performance, and 2) the proposed regularisation term enables long-term predictions in the presence of noisy and irregularly sampled data.\nThe concept of a binary knowledge mask is well motivated. However, domain expertise is often subject to at least some level of uncertainty. However, it is not clear from the discussions in the paper if and how this uncertainty is handled within proposed model.\n\nRegarding the regularisation term, the paper remarks on p. 5 that “this term is implemented as a Euclidean distance penalty between the sample $z(ti)$ from the prior distribution and the sample $z^{\\ast} (ti)$ from the posterior distribution.” Considering that the we don’t know the space in which the latent states are located, how do we know that the Euclidean distance is an appropriate distance? The authors argue that the experimental results and ablations validate this choice. While the experiments provide convincing results for the model itself, I couldn’t find any ablations in the paper or the appendix that focused on the specific choice of the metric that is used to implement the regularization term. I would have liked to see a comparison between Euclidean distance against alternative distance measures/metrics.", "methods_and_evaluation_criteria": "The experimental results clearly evidence the improvements in performance compared to the state of the art. I particularly appreciate the distinction between the interpolation and extrapolation tasks that provide insight into the model’s ability to generalise. The ablation studies in Appendix H provide additional insight how the two proposed components (Phy-SSM unit; regularisation term) contribute to the overall performance of the system.  \n\nThat said, the results in Table 7 seem to suggest that the performance of the Phy-SSM unit (consisting of a 4-layer MLP, 9 SSM layers, and a 4-layer MLP, see p. 17) is comparable to a unit that consists only of MLPs for the interpolation task, and leads to small improvements for the extrapolation task. However, it is unclear at what cost these improvements in extrapolation are achieved. To this end, I feel that the paper would benefit from a more thorough treatment of the computational cost of the proposed Phy-SSM module. To ensure a balanced discussion of the achievements and limitations of the proposed model, I would also recommend to move the ablation study from the appendix into the main body of the paper.\n\nMoreover, I am surprised that a unit involving a simple MLP outperforms a unit involving standard SSMs (as opposed to Phy-SSMs) by a considerable margin for the extrapolation task. How much of this improvement can be attributed to the regularisation term? It would be helpful to include one more row in the table to replaces the Phy-SSM unit with the MLP but does not incorporate regularisation.", "theoretical_claims": "See “Claims & evidence”", "experimental_designs_or_analyses": "See “Methods and Evaluation”", "relation_to_broader_scientific_literature": "In my opinion, the novel contribution of the paper is the extension of SSMs to extrapolation. In my opinion, the results – particularly the trajectory plots in Appendix G – clearly highlight the shortcomings of S5 for extrapolation, and showcase the significant improvements that can be achieved through regularisation and incorporation of prior knowledge.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper well written and provides insightful examples and diagrams to help illustrate the proposed model. Detailed information about the experimental settings and datasets are provided in the appendix. I particularly appreciated the information in Appendix D.2 that details prior knowledge / models of the systems used for the experimental evaluation.  In addition, Appendix G provides convincing plots of the trajectory estimates for different baseline models.", "other_comments_or_suggestions": "- P. 4,  left column, line 207: “approximated posterior $z(t_i)$” -> “approximate posterior of $z(t_i)$?\n- P. 4, right column, line 214: “influence of control inputs is often known” – do you mean “unknown”?\n- Section 4.3 feels out of place. I would suggest to swap 4.2 and 4.3\n- P. 28, line 1490: “SMM” -> “SSM”", "questions_for_authors": "1.\tHow is uncertainty in the “known dynamics” handled within the proposed model considering that a binary knowledge mask is applied to separate frozen and learnable elements of the state matrix?\n2.\tHow does the Euclidean distance for regularisation term compare to alternative metrics?\n3.\tWhat is the computational cost of the Phy-SSM module (e.g., compared to the standard SSM and MLP in Appendix H)?\n4.\tIn Section 5, how was the Phy-SSM module initialised? How was S5 initialised?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YLMALIb7VN", "reviewer": "Reviewer_gB8T", "summary": "This paper addresses the problem of dynamic forecasting with noisy and irregularly sampled data. A model is proposed that 1) a physics-based SSM is applied to integrate partial physics knowledge and 2) a physics state regularization is used to constrain the latent states with noisy and irregularly sampled data. Empirical results show improved performance of the proposed model on interpolation and extrapolation tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "## Response to Reviewer gB8T\n**Q 4.1**: About clarifying the difference between our method and existing hybrid modeling approaches [1-3] that address incomplete physics knowledge.\n\n**A 4.1**: We have discussed the difference between our model and hybrid modeling approaches as follows. The models in [1–3] are all physics-based NODE methods, which suffer from a shared limitation: difficulty in modeling nonlinear and time-variant systems over long time horizons due to their heavy reliance on initial conditions. This limits their ability to capture long-term sequence dependencies.\n\nPI-VAE [2] is already included as a baseline in **Tables 1–4** of our submission, and our method outperforms it. We will include references [1] and [3] in the related work section of the revised version.\n\n>[1] Yin et al. Augmenting physical models with deep networks for complex dynamics forecasting. JSTAT 2021\n>\n>[2] Takeishi et al. Physics-integrated variational autoencoders for robust and interpretable generative modeling. NIPS 2021\n>\n>[3] Wehenkel et al. Robust hybrid learning with expert augmentation. TMLR 2023\n\n**Q 4.2**: About (i) comparing with baselines like ODE2VAE; (ii) NODE limitations on initialization sensitivity.\n\n**A 4.2**: We have included ODE2VAE [4] as a baseline in our experiments. The results are provided in **Tables 1-4** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)]. While ODE2VAE introduces uncertainty modeling to alleviate the sensitivity to initial conditions, it still lacks an effective mechanism to dynamically refine predictions based on later observations. As a result, its performance on long-term forecasting remains limited.\n\nRegarding the discussion of initialization in Section 2, we refer specifically to the sensitivity of NODE-based methods to initial conditions $x(t_0)$, not to the initialization of network parameters. This is due to the fact that NODEs solve initial value problems (IVPs), where the trajectory is determined by both the initial state and the learned vector field. In contrast, our Phy-SSM adopts a dynamical VAE framework that refines its predictions over time using the posterior from the preceding time step. This mechanism effectively mitigates errors caused by inaccurate initial conditions.\n\nA motivating example that visually illustrates this improvement is provided in Fig. 2, Appendix A.\n\n>[4] Yildiz et al. Ode2vae. NIPS 2019\n\n**Q 4.3**: About how Phy-SSM handles continuous dynamics and the definition of $\\psi(z)$.\n\n**A 4.3**: The Phy-SSM unit learns continuous dynamics through the parameterized matrices $A_{unk}$ and $B_{unk}$ as showed in Eq. (8), as illustrated in S5 [5]. \n\n$\\psi(z)$ denotes additional extended state terms, which can include nonlinear functions or constants—such as $\\sin{({z})}$, $\\cos{({z})}$ and 1. This augmentation enables the representation of certain nonlinear systems in a linear state-space form. Please refer to page 4, right column, lines 192–200 for the detailed explanation.\n\n>[5] Smith et al. S5. ICLR 2023\n\n**Q 4.4**: About defining the knowledge mask when physics is fully unknown.\n\n**A 4.4**: If physics is unknown, we treat the entire system as unknown and set all mask entries to 1. In this case, our method reduces to a data-driven SSM that learns all dynamics from data without explicit constraints.\n\n**Q 4.5**: About elaborating on why the L2 norm of the latent states from the prior and posterior distribution is used.\n\n**A 4.5**: We use L2-norm for its efficiency and strong empirical performance. We have conducted comparisons with alternative metrics(e.g., Chebyshev, cosine). The results are provided in **Table 5** [[link](https://anonymous.4open.science/r/ICMLExp-6D54/Rebuttal_exp.pdf)], where Euclidean distance (L2-norm) performs best in extrapolation tasks. \n\n\n**Q 4.6**: About the irregularity details of the three real-world datasets used in the experiments.\n\n**A 4.6**: Appendix E in our submission provides the details. Drone data is high-frequency and irregularly sampled, recorded at nearly 1010 Hz (minimum: 573.05 Hz, maximum: 1915.86 Hz); COVID-19 contains 10% missing daily records; vehicle dataset includes 5% missing agent observations. We will add clearer descriptions in the revised version.\n\n**Q 4.7**: About the seemingly marginal contribution of the physics state regularization term and the request for a regularization-only comparison.\n\n**A 4.7**: We would like to emphasize that the physics state regularization yields approximately a 10% relative improvement in extrapolation performance. It plays a crucial role in guiding the model toward learning more generalized physically representations—particularly under noisy and irregular data conditions.\n\nIt is important to note that the regularization term cannot be used independently, as it penalizes the distance between the posterior (output by the sequential encoder) and the prior physics-based latent states predicted by the Phy-SSM unit.", "claims_and_evidence": "1. The challenge of noisy and irregularly sampled data in long-term dynamics forecasting is critical.\n2. In Section 2 (ii) the authors stated that the existing works did not consider the infeasibility of obtaining complete physics knowledge. However, there is an existing domain of hybrid modeling that aims to solve this problem [1-3]. The authors should further discuss the difference between the proposed model compared to hybrid modeling.\n3. Section 2 also mentioned the limitation of NODE on nonlinear and time-variant systems. However, there have been works such as ODE2VAE [4] for such complex systems. The authors should check these works for better comparison. Also, the authors stated that the initialization of NODE-based models is critical. Could the authors compare the initialization in the three experimental settings to show how that is improved by the proposed model?\n\n[1] Yin, Yuan, et al. \"Augmenting physical models with deep networks for complex dynamics forecasting.\" Journal of Statistical Mechanics: Theory and Experiment 2021.12 (2021): 124012.\n\n[2] Takeishi, Naoya, and Alexandros Kalousis. \"Physics-integrated variational autoencoders for robust and interpretable generative modeling.\" Advances in Neural Information Processing Systems 34 (2021): 14809-14821.\n\n[3] Wehenkel, Antoine, et al. \"Robust hybrid learning with expert augmentation.\" arXiv preprint arXiv:2202.03881 (2022).\n\n[4] Yildiz, Cagatay, Markus Heinonen, and Harri Lahdesmaki. \"Ode2vae: Deep generative second order odes with bayesian neural networks.\" Advances in Neural Information Processing Systems 32 (2019).", "methods_and_evaluation_criteria": "1. Based on Eq 7 and Eq 8, the proposed Phy-SSM unit is similar to an RNN-structured sequential model. How does this unit process the continuous dynamics? Also, how is the function $\\psi(z)$ defined?\n2. Eq 9 introduced the knowledge mask mechanism. For real-world systems where the physics is usually unknown, it is not feasible to explicitly write out the mask as in the example in Section 4.2. How does the proposed method deal with such a problem?\n3. The physics regularization in Eq 12 is supposed to be a major contribution as stated in Section 1. The authors should elaborate on why the L2 norm of the latent states from the prior and posterior distribution is used.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "1. The authors used three real-world datasets in experiments. How irregular are the three datasets? Could the authors provide more details?\n2. From the ablation study it seems the physics state regularization has a marginal contribution to the model improvement. Could the authors discuss this phenomenon? Also, could the authors compare the model with regularization only?", "relation_to_broader_scientific_literature": "Please find my comments above.", "essential_references_not_discussed": "Please find my comments above.", "other_strengths_and_weaknesses": "Please find my comments above.", "other_comments_or_suggestions": "Please find my comments above.", "questions_for_authors": "Please find my questions above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes an approach to integrate (partially-known) physical laws with a deep state-space model. 
The proposed method Phy-SSM is benchmarked across three diverse application domains against state-of-the-art methods (with diverse
architectures, e.g. deep SSMs, RNNs, a Transformer, etc.) for long-term forecasting in physical systems. The reviewers initially 
expressed concerns about the lack of a Transformer-based model, but this was adressed during the rebuttal (Contiformer was added), 
and the lack of a physical system with a well-known governing equation. Overall the reviewers found the paper to be well-written
and containing helpful illustrations and explanations. One concern raised in the reviews and shared by me is that presenting the
method as "general-purpose" is potentially overstating what is demonstrated experimentally. I urge the authors to tone down
the language used to describe their approach and to release the code as promised. Overall I believe the positives outweigh the
negatives and i recommend this work be accepted.
Decision: Accept (poster)