Phy-SSM: Physics-Enhanced State Space Model

• Phy-SSM is a physics-enhanced state space model that integrates known physics and learnable unknown dynamics for improved long-term forecasting.
• It employs a gray-box decomposition and physics state regularization to align latent states with physical dynamics in noisy, irregular settings.
• Empirical results on vehicle, drone, and COVID-19 datasets demonstrate that Phy-SSM outperforms baselines in both interpolation and extrapolation tasks.

Phy-SSM is a physics-enhanced state space model for long-term dynamics forecasting in complex environments with noisy and irregularly sampled observations, designed for both interpolation and extrapolation when only partial physics knowledge of the underlying system is available. Its central construction is a gray-box decomposition of latent dynamics into known and unknown state matrices inside a deep state space model, combined with a physics state regularization term that aligns latent states inferred from observations with latent states predicted by the physics-aware prior dynamics. The method was introduced for vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting, where it was reported to outperform a range of baselines in long-term interpolation and extrapolation tasks (Wang et al., 14 Jul 2025).

1. Definition, scope, and conceptual position

Phy-SSM, short for Physics-Enhanced State Space Model, addresses long-term forecasting in settings where data are noisy, observations are irregularly spaced in time, and the system dynamics are only partially known. The method is motivated by the observation that state space models can capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge can improve generalization ability (Wang et al., 14 Jul 2025).

The forecasting setting is formulated through an observation model

$\bm{x}(t) = \bm{g} \left( \bm{z}(t) \right),$

where $\bm{x}(t)$ is the observation and $\bm{z}(t)$ is the latent or full system state, together with latent dynamics

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$

where $\bm{u}(t)$ is a control input. Inputs are observation sequences $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$ with $0=t_0<t_1<\dots<t_n=T$, and control sequences $$[\bm{u}(t_0),\cdots,\bm{u}(t_n),\cdots,\bm{u}(t_{n+l})]$$. Outputs are predicted observations for interpolation within the observed window and for extrapolation beyond it (Wang et al., 14 Jul 2025).

Within the broader literature on state space models, Phy-SSM is positioned between purely data-driven sequence models and fully specified physics-based models. Purely data-driven state space models can fit observed trajectories and capture long-range dependencies, but may produce physics-inconsistent extrapolations. Physics-enhanced Neural ODE-style methods can inject prior knowledge, but were argued to rely heavily on initial conditions and to lack a strong mechanism for continually refining latent dynamics using subsequent observations under noisy and irregular sampling. Phy-SSM addresses this by embedding partial physics directly inside the latent transition mechanism rather than only in the loss (Wang et al., 14 Jul 2025).

A recurrent source of confusion is the abbreviation itself. In a different domain, “Phy-SSM” has also been used informally for phylogenetic stochastic mapping without matrix exponentiation, an exact MCMC method for conditional CTMC histories on phylogenies; that usage is unrelated to the physics-enhanced forecasting model described here (Irvahn et al., 2014).

2. Formal structure of the latent dynamics

The defining step in Phy-SSM is the decomposition of partially known dynamics into known and unknown components:

$\dv{\bm{z}(t)}{t} = \fB(\bm{z}(t), \bm{u}(t)) = \fB_{\mathrm{knw}}(\bm{z}(t), \bm{u}(t)) + \fB_{\mathrm{unk}}(\bm{z}(t), \bm{u}(t)).$

To embed this inside a state space model, the latent state is lifted to an extended state

$$\bar{\bm{z}} = [\bm{z}^\top, \bm{\psi}(\bm{z})^\top]^\top \in \mathbb{R}^{d_{\bar z}},$$

where $\bm{x}(t)$0 may contain nonlinear terms or constants. This yields the continuous-time linear-operator form

$\bm{x}(t)$1

The interpretation is explicit: $\bm{x}(t)$2 encodes known physical dynamics, $\bm{x}(t)$3 encodes unknown latent dynamics learned from data, and $\bm{x}(t)$4 encodes generally unknown control effects (Wang et al., 14 Jul 2025).

This formulation is neither a purely black-box latent recurrence nor a fully specified mechanistic model. A plausible implication is that Phy-SSM should be understood as a structured latent dynamical system whose inductive bias is determined by the support and values of the known matrices, while the residual degrees of freedom are delegated to learnable state space components.

The paper further relates the state lifting to SINDy/SINDYc-style augmentation: nonlinear dynamics can become more linearly representable in an expanded coordinate system. That connection is operational rather than merely interpretive, because the extended state is the object propagated by the Phy-SSM transition model (Wang et al., 14 Jul 2025).

3. Architecture and inference pipeline

The model consists of three main components: a sequential encoder, a Phy-SSM unit, and a decoder (Wang et al., 14 Jul 2025).

The sequential encoder estimates a posterior over latent states from current observations and memory:

$\bm{x}(t)$5

with

$\bm{x}(t)$6

The trajectory posterior is approximated as

$\bm{x}(t)$7

This factorization is introduced to reduce cost while preserving a sequential dependence on preceding observations (Wang et al., 14 Jul 2025).

Unknown continuous dynamics are learned through multi-layer structured state space models that produce $\bm{x}(t)$8 and $\bm{x}(t)$9 as functions of time, state, and control. These are then filtered by binary knowledge masks

$\bm{z}(t)$0

to obtain

$\bm{z}(t)$1

A mask entry of $\bm{z}(t)$2 means the term is learnable; a mask entry of $\bm{z}(t)$3 blocks learning so that known physics remains fixed (Wang et al., 14 Jul 2025).

The continuous dynamics are discretized with the bilinear method. Using the resulting discrete matrices $\bm{z}(t)$4 and $\bm{z}(t)$5, the transition becomes

$\bm{z}(t)$6

This makes the model continuous-time in formulation but discrete-time in computation, with the discretization depending on the time step and therefore compatible with irregular sampling (Wang et al., 14 Jul 2025).

The decoder maps latent states back to observations, and the prior over latent states is itself physics-based rather than standard normal:

$\bm{z}(t)$7

During interpolation, the continuous Phy-SSM unit refines predictions using posterior information from preceding steps inside the observation window. During extrapolation, the model rolls forward autoregressively from the last inferred latent state using the learned physics-aware dynamics and control inputs (Wang et al., 14 Jul 2025).

4. Physics integration and state regularization

The most distinctive feature of Phy-SSM is the way partial physics enters the architecture. Known physical structure is encoded directly in $\bm{z}(t)$8, while unknown structure is learned only on the entries permitted by the masks. This means the model does not simply penalize physics violations after the fact; it restricts the learnable transition operator at the level of support. The paper describes this as a hard-constraint mechanism that uses disjoint support for known and unknown matrices (Wang et al., 14 Jul 2025).

The training objective combines a VAE loss with a physics state regularization term:

$\bm{z}(t)$9

where

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$0

with

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$1

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$2

and

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$3

Here, $\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$4 is sampled from the prior and $\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$5 is sampled from the posterior (Wang et al., 14 Jul 2025).

The stated role of $\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$6 is twofold. First, it constrains encoder-inferred latent states to adhere to physical dynamics. Second, it helps the Phy-SSM unit learn more accurate unknown dynamics aligned with the whole trajectory, which was reported to improve long-term extrapolation. Appendix experiments compare Chebyshev, cosine, and Euclidean distances for this regularizer, and Euclidean distance performed best on extrapolation (Wang et al., 14 Jul 2025).

The paper’s decomposition examples make the mechanism concrete. For a pendulum with unknown friction and unknown length,

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$7

the state is augmented with

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$8

giving

$\dv{\bm{z}(t)}{t} = \fB \left( \bm{z}(t), \bm{u}(t) \right),$9

The known and unknown parts are then separated so that only the missing row is learned:

$\bm{u}(t)$0

with mask

$\bm{u}(t)$1

The COVID-19 example uses the SIR structure

$\bm{u}(t)$2

with unknown time-varying $\bm{u}(t)$3 and $\bm{u}(t)$4, but known structural dependencies (Wang et al., 14 Jul 2025).

5. Theoretical properties, training, and implementation

Phy-SSM includes a uniqueness statement for the decomposition. The paper states:

Proposition 1 (Uniqueness). For a dynamical system in the form of Eq. $\bm{u}(t)$5, if it can be reformulated as Eq. $\bm{u}(t)$6, the decomposition in Eq. $\bm{u}(t)$7 that minimizes Eq. $\bm{u}(t)$8 is unique. (Wang et al., 14 Jul 2025)

The argument relies on two assumptions: an encoder/decoder approximation assumption based on universal approximation, and the assumption that there exists one and only one underlying dynamics $\bm{u}(t)$9 minimizing the loss. The key structural condition is that $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$0 and $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$1 have disjoint support. Elementwise, the paper writes

$[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$2

$[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$3

so that

$[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$4

This does not amount to a full nonparametric identifiability theorem, but it formalizes the architectural claim that known and unknown terms do not compete for the same support (Wang et al., 14 Jul 2025).

Training is end-to-end in PyTorch on a server with 4 NVIDIA A6000 GPUs. The appendix reports application-specific optimization settings. For drone state prediction, the optimizer is Adam with learning rate $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$5, 20 epochs, and batch size 64. For COVID-19 forecasting, it is Adam with learning rate $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$6, 400 epochs, and batch size 32. For vehicle motion prediction, it is AdamW with cosine one-cycle schedule, 80 epochs, and batch size 64. For the pendulum experiment, it is Adam with learning rate $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$7, 150 epochs, and batch size 64 (Wang et al., 14 Jul 2025).

The architecture also varies by task. On drone data, the encoder is a 4-layer MLP with hidden size 200 plus a 5-layer SSM with 128 hidden states; the unknown dynamics module is a 4-layer SSM with 128 hidden units per layer; and the decoder is a 4-layer MLP with hidden size 200. On COVID-19 data, the encoder is a 3-layer MLP with hidden size 200 plus a 4-layer SSM with 128 hidden states; the unknown dynamics module is a 3-layer SSM with 128 hidden units per layer; and the decoder is a 2-layer MLP with hidden size 200. On vehicle motion prediction, the encoder is a 2-layer SSM with 256 hidden states, the unknown dynamics module is a 4-layer SSM with 256 hidden units per layer, and the decoder is a 2-layer MLP with 256 hidden units; control inputs are embedded by an off-the-shelf scene encoder and fused by cross-attention (Wang et al., 14 Jul 2025).

6. Empirical evaluation, results, and interpretation

The reported experiments cover three real-world applications: vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting (Wang et al., 14 Jul 2025).

For vehicle motion prediction, the dataset is nuScenes. The model uses the first 2 seconds as input, predicts the next 5 seconds during training, and is tested with an extended 6-second horizon. The setup includes 5% missing agent observations. Metrics are ADE, FDE, Speed Error, Acceleration Error, and Jerk Error. In-domain extrapolation results for Phy-SSM are ADE 1.884, FDE 5.100, Speed Error 1.336, Acceleration Error 2.399, and Jerk Error 1.884. Out-of-domain extrapolation results are ADE 6.206, FDE 7.197, Speed Error 2.398, Acceleration Error 2.043, and Jerk Error 1.548 (Wang et al., 14 Jul 2025).

For drone state prediction, the dataset is a real quadrotor dataset recorded at irregular high frequency around 1010 Hz, with minimum 573.05 Hz and maximum 1915.86 Hz. The task uses 800 timesteps to predict the next 200 timesteps. Metrics are MAE and MSE. On interpolation, the best Phy-SSM results are MAE 1.002 $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$8 0.034 and MSE 0.222 $[\bm{x}(t_0),\bm{x}(t_1),\cdots,\bm{x}(t_n)]$9 0.020, both reported with scaling $0=t_0<t_1<\dots<t_n=T$0. On extrapolation, Phy-SSM reports MAE 2.733 $0=t_0<t_1<\dots<t_n=T$1 0.059 and MSE 1.798 $0=t_0<t_1<\dots<t_n=T$2 0.079 (Wang et al., 14 Jul 2025).

For COVID-19 forecasting, the data are from JHU/Covsirphy, with 10% daily records randomly dropped. The task uses 160 irregular samples to predict the next 80 future days, with training on some countries and validation/test on unseen countries. Metrics are MAE and MSE. On interpolation, Phy-SSM reports MAE 0.795 $0=t_0<t_1<\dots<t_n=T$3 0.208 $0=t_0<t_1<\dots<t_n=T$4 and MSE 1.032 $0=t_0<t_1<\dots<t_n=T$5 0.538 $0=t_0<t_1<\dots<t_n=T$6. On extrapolation, it reports MAE 1.998 $0=t_0<t_1<\dots<t_n=T$7 0.753 and MSE 0.692 $0=t_0<t_1<\dots<t_n=T$8 0.486 (Wang et al., 14 Jul 2025).

The baseline set includes Latent ODE with RNN Encoder, Latent ODE with ODE-RNN Encoder, ContiFormer, S5, GOKU, PI-VAE, SDVAE, and ODE2VAE; vehicle experiments additionally include Wayformer, AutoBot, and G2LTraj. The paper’s main empirical claim is that Phy-SSM performs particularly strongly on long-term extrapolation, with the most dramatic reported gains appearing on COVID-19 forecasting and with substantial improvements over baseline extrapolation on drone data (Wang et al., 14 Jul 2025).

The ablation study on drone data isolates two components: the Phy-SSM unit and the physics state regularization. Removing both yields interpolation MAE 1.059, extrapolation MAE 8.426, and extrapolation MSE 17.333. Adding the Phy-SSM unit without regularization gives interpolation MAE 0.927, extrapolation MAE 3.008, and extrapolation MSE 2.176. The full model gives interpolation MAE 1.002, extrapolation MAE 2.733, and extrapolation MSE 1.798. This suggests that the Phy-SSM unit is the major source of extrapolation improvement, while the regularizer trades a small amount of interpolation fit for better long-term generalization (Wang et al., 14 Jul 2025).

A broader implication, supported by the paper’s framing but still interpretive, is that Phy-SSM is best suited to gray-box forecasting regimes: systems with meaningful but incomplete structural knowledge, irregular sampling, noisy observations, and a strong need for extrapolation rather than only one-step prediction.

7. Relation to the broader SSM literature and limitations

Phy-SSM belongs to the contemporary state space model family rather than to the anatomical statistical shape modeling literature that also uses the acronym “SSM.” In the sequence-modeling sense, state space models have been studied as efficient long-range sequence architectures with favorable scaling, and later systems work has shown that pure SSMs can process much longer contexts than comparable Transformers on consumer hardware, with custom SSM kernels becoming dominant runtime bottlenecks at long context (Mitra et al., 16 Jul 2025). This broader context helps situate Phy-SSM: its novelty is not the use of state space models alone, but the explicit insertion of partially known physics into the state transition structure (Wang et al., 14 Jul 2025).

The paper also differentiates Phy-SSM from other physics-informed model classes. Unlike PINN-style formulations, it does not encode physics primarily through residual penalties in the loss. Unlike Hamiltonian, Lagrangian, or symplectic methods, it does not require a specialized mechanics formalism. Unlike standard Neural ODE approaches, it uses a structured state space recurrence with memory together with a posterior-prior alignment mechanism. This suggests that its intended domain is broad gray-box dynamical modeling rather than narrowly mechanistic scientific simulation (Wang et al., 14 Jul 2025).

The main limitations are explicit or strongly implied. The system must be reformulable in the lifted linear-SSM form. A manually designed knowledge mask is required. The method assumes that some meaningful partial physics is available. The uniqueness result depends on strong assumptions, especially uniqueness of the underlying loss-minimizing dynamics. The paper also notes that if the provided physics is inaccurate or the mask is poorly specified, the inductive bias may become harmful rather than beneficial. These are structural limitations of the method rather than incidental training issues (Wang et al., 14 Jul 2025).

In summary, Phy-SSM is a gray-box forecasting framework that integrates partial physics into deep state space models through support-constrained dynamics decomposition and latent-state alignment. Its distinctive contribution is architectural: known and unknown dynamics are separated inside the transition operator itself, rather than merely reconciled by post hoc regularization. Its reported empirical profile is strongest on irregular, noisy, long-horizon forecasting tasks where extrapolation quality is the central criterion (Wang et al., 14 Jul 2025).