---
title: 'OR-GSSM: Physics-Informed Recurrent SSM'
url: https://www.emergentmind.com/topics/output-recurrent-gated-state-space-model-or-gssm
type: topic
---

# OR-GSSM: Physics-Informed Recurrent SSM

The **Output-Recurrent Gated State Space Model (OR-GSSM)** is a physics-informed recurrent architecture introduced for **complex multiphase flows modeling and uncertainty quantification of exhaust vehicles during motion**. It is constructed from a **state-space formulation of the gas-liquid Navier-Stokes equations** obtained by **applying semigroup theory and Galerkin projection**, and it combines that formulation with a **Gated State Space Transition (GSST) unit** and an **output recursion mechanism** designed to match the numerical solution characteristics of state-space equations. In the reported experiments, OR-GSSM is applied to underwater cone-head and water-exit hemisphere-head vehicles, where it is used to model velocity, pressure, and volume-fraction fields and to quantify predictive uncertainty [2508.00588].

## 1. State-space formulation and physical variables

OR-GSSM is built on a discrete linear state-space model with process and measurement noise,
$$
x_{t+1}=A(\theta)\,x_t + B(\theta)\,u_t + w_t,\qquad
y_t=C\,x_t + v_t,
$$
where $x_t\in\mathbb{R}^{n_x}$ is the hidden state, $u_t\in\mathbb{R}^{n_u}$ the control or input, $y_t\in\mathbb{R}^{n_y}$ the observed output, and $w_t$, $v_t$ are zero-mean process and measurement noises.

In the multiphase-flow setting around an exhaust vehicle, the paper defines the state, input, and output as follows. The state vector is
$$
x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},
$$
which collects the time-coefficients of the Galerkin-modal expansions of velocity $U$ and liquid-volume fraction $\alpha$. Pressure is **not** included directly in the state; instead, it is treated as a **Lagrange multiplier** and recovered via a **discrete Poisson solve**. The input is
$$
u_t := \{T_{in}(t)\}\in\mathbb{R}^{n_u},
$$
representing the dimensionless external parameters—**environmental pressure, navigation speed, exhaust flow rate**—derived via **Buckingham’s $\Pi$-theorem**. The output is
$$
y_t := \{T_{out}(t)\}\in\mathbb{R}^{n_y},
$$
namely the dimensionless physical fields—**velocity components, pressure, volume fraction**—on the Eulerian grid [2508.00588].

This formulation explicitly characterizes the dynamic coupling evolution between the **velocity**, **pressure**, and **volume fraction** fields. A common misunderstanding would be to read OR-GSSM as a purely data-driven recurrent surrogate detached from continuum mechanics; the paper instead presents it as a model whose backbone is derived from a two-phase Navier-Stokes state-space construction.

## 2. Gated State Space Transition unit

The core recurrent block is the **Gated State Space Transition (GSST) unit**. At time step $t$, the GSST takes the new external input $x_t$ (identified in the detailed exposition with $T_{in}$) and the previous hidden state $h_{t-1}$ and computes
$$
Z_t=\sigma\bigl(W_z\,[h_{t-1},x_t]+b_z\bigr),
$$
$$
\Delta_t=\mathrm{softplus}\bigl(\mathrm{AvgPool}(W_\Delta \ast x_t)\bigr)>0,
$$
$$
A_t=\exp\bigl(-\Delta_t\,\hat A\bigr),\qquad
\hat A=\exp(W_A)\in\mathbb{R}^{d\times d},
$$
$$
B_t=\Delta_t\,f_B(W_B\ast x_t),
$$
$$
\tilde h_t=A_t\,h_{t-1}+B_t\,f_n(W_n\ast x_t),
$$
$$
h_t=(1-Z_t)\odot h_{t-1}+Z_t\odot\tanh\bigl(\tilde h_t+f_x(W_x\ast x_t)\bigr).
$$

Here, $\ast$ denotes a spatial convolution, $\sigma(\cdot)$ is the element-wise sigmoid, $\mathrm{softplus}(u)=\ln(1+e^u)$ ensures $\Delta_t>0$, the tensors $W_z,b_z,W_\Delta,W_A,W_B,W_n,W_x$ are trainable convolution kernels, the functions $f_B,f_n,f_x$ are pointwise nonlinearities such as **Swish**, and $d=d_{\text{hidden}}$ is the hidden dimension.

The paper assigns specific state-space interpretations to these quantities. By design, $A_t$ plays the role of $\exp(\Delta_t A)$ in a **zero-order-hold discretization** of the continuous SSM, while $B_t\approx \Delta_t B$ when $\Delta_t$ is small. The gate $Z_t$, described as inspired by **GRU**, adaptively mixes old and new information [2508.00588].

This architecture is therefore not merely a gated RNN cell with renamed variables. The explicit parameterization of $A_t$, $B_t$, and $\Delta_t$ is presented as the mechanism through which OR-GSSM acquires **adaptive timescales**, **physical interpretability**, and **computational efficiency**.

## 3. Output recurrence and numerical consistency

A defining feature of OR-GSSM is its **output recursion mechanism**. Instead of using teacher forcing, the model feeds its own predicted output back as part of the next input, in a way the paper describes as matching the numerical solution of a state-space ODE. The recurrence is written as
$$
D_t=E_{\rm enc}(T_{in,t}),
$$
$$
h_t=\mathrm{GSST}(h_{t-1},T_{in,t},D_t),
$$
$$
\hat y_t=C\,h_t\odot s,
$$
where $s$ is a spatial mask that is zero inside the vehicle and one in the fluid.

The key training rule is explicit: **the model always uses $\hat y_{t-1}$, not the ground truth, as part of the next input**. The paper frames this choice as a way to avoid the **training-inference pattern mismatch** associated with **teacher forcing** and **scheduled sampling**. It further states that the gradient of the MSE loss with respect to $\theta$ acquires extra terms from the recurrence path, and that these terms **dynamically constrain parameter updates** and **mitigate long-term error accumulation** [2508.00588].

This distinction is central to the model’s identity. OR-GSSM is not simply “state-space plus autoregression” in a generic sense; its output recurrence is motivated by **numerical consistency** with state-space evolution. The paper also attributes to this mechanism **more stable training**, **better generalization**, and **higher prediction accuracy** than teacher forcing and scheduled sampling.

## 4. Optimization, normalization, and inference procedure

Training uses a loss consisting of an MSE term over sequences and time steps together with $\lambda\|\theta\|_2^2$ regularization. The regularization coefficient is reported as
$$
\lambda \approx 10^{-4}\ldots 10^{-2}.
$$
Optimization uses **Adam** with an initial learning rate of $10^{-3}$, halved every **500 epochs**, for a total of **2 500 epochs**. The **batch size** is **5 sequences**. The reported architecture uses **hidden dimension $d_{\text{hidden}}=128$**, **4 transposed-conv layers in $E_{\rm enc}$**, and a **spatial grid of $128\times 128$** [2508.00588].

The paper emphasizes that both input and output are **normalized (dimensionless)**, and states that this ensures **numerical stability**, specifically noting the absence of **gradient explosion/vanishing**. Inference is performed by iterating the recurrence forward over the input sequence $T_{in}$, recycling $\hat y_{t-1}$ at each step.

The training and inference loops given in the detailed exposition are operationally simple: initialize the parameters
$$
\theta=(W_z,b_z,W_\Delta,W_A,W_B,W_n,W_x,E_{\rm enc},C),
$$
set the initial hidden state to zero, encode the input, propagate with GSST, form the masked output $(Ch_t)\odot s$, accumulate the sequence loss, and update $\theta$ with Adam. A plausible implication is that the model is intended to preserve the deployment-time recurrence during optimization rather than approximating it indirectly through a teacher-forced surrogate objective.

## 5. Empirical validation in underwater and water-exit exhaust flows

The paper validates OR-GSSM on two exhaust-vehicle scenarios: an **underwater cone-head vehicle with shoulder exhaust** and a **water-exit hemisphere-head vehicle**. In the underwater cone-head case, a **2D axisymmetric cone** with **$D=40$ mm**, **$L=290$ mm**, and **exhaust seam $w=1$ mm at 19 mm** is simulated using **high-fidelity CFD (VOF + $k$–$\omega$ SST)**. Metrics are reported over **600 test samples** as time-averaged $L_2$-norm relative error in percent, together with per-iteration runtime on an **NVIDIA V100**.

| Model | Error profile $(\|u\|_{\rm err}, p_{\rm err}, \alpha_{\rm err}, p_{{veh},\rm err})$ | Runtime |
|---|---|---|
| OR-ConvLSTM | 6.08, 1.69, 13.59, 2.67 | 1.42 s |
| OR-ConvGRU | 6.05, 1.70, 13.55, 2.73 | 1.17 s |
| OR-GSSM | 6.02, 1.61, 13.55, 2.63 | 0.98 s |

The paper states that OR-GSSM is both more accurate and faster than the OR-ConvLSTM and OR-ConvGRU baselines, attributing this to **parallel matrix-exponential and convolutional operations** rather than **sequential multi-gate RNN updates** [2508.00588].

For the underwater cone-head validation, the computational domain is **$[-0.4,0.01]\times[-0.05,0.05]$ m** on a **$128\times128$ grid** with **velocity inlet**, **pressure outlet**, **symmetry axis**, and **no-slip walls**. Training uses **100 CFD runs** over
$$
P_e\sim U(45,90)\ {\rm kPa},\quad
V_e\sim U(5,11)\ {\rm m/s},\quad
Q_e\sim U(1.2,2.6)\ {\rm g/s}.
$$
Testing and uncertainty quantification use **600 runs** at
$$
P_e=66.2\ {\rm kPa}\pm 10\%,\quad
V_e=8\ {\rm m/s}\pm 10\%,\quad
Q_e=1.92\ {\rm g/s}\pm 10\%.
$$
The reported captured phenomena are **gas-phase region expansion**, **formation of gas-liquid mixing layer**, **bubble breakup and shedding**, **backflow-jet generation**, and the **spatial distribution of surface pressure plateau in the gas region and peak in backflow region**.

For the water-exit hemisphere-head vehicle, the geometry is specified as **$D=2$ m**, **$L=13$ m**, **seam at 2.5 m**, **internal cavity $V=7.69\ {\rm m}^3$**, and **$m=59\,000$ kg**. The domain is **axisymmetric**, extending **30 m below/above free surface**, with mesh refinement near the interface. The physical modeling uses a **Mixture model for exhaust-flow coupling** and **$k$–$\omega$ SST for hydrodynamics**. Training uses **100 runs** over
$$
P_e\sim U(300,400)\ {\rm kPa},\quad
V_e\sim U(15,25)\ {\rm m/s},
$$
while testing and uncertainty quantification use **200 runs** at
$$
P_e=350\ {\rm kPa}\pm 10\%,\quad
V_e=20\ {\rm m/s}\pm 10\%.
$$
The captured phenomena include **fully-coupled underwater→surface→air-exposure stages**, **splash-up and cavity detachment**, **dynamic pressure release at the head upon exit**, and **residual cavity pressures**.

## 6. Uncertainty quantification, computational profile, and positioning

The uncertainty quantification results are reported separately for both validation regimes. In the underwater cone-head case, the **mean flow-fields match CFD**, the **largest standard deviation** appears in the **mixing/backflow regions**, and **surface-pressure predictions lie within $\pm 1\sigma$ confidence bands**. In the water-exit hemisphere-head case, statistics at fixed depths
$$
h=16.2,\ 12.2,\ 6.2,\ 0.2\ {\rm m}
$$
show **larger uncertainty near the separating interface and backflow region** [2508.00588].

The computational profile given in the paper identifies the per-step cost as dominated by **convolutional blocks**
$$
O(N_xN_y\cdot d_{\text{hidden}}\cdot k^2)
$$
and **channel-wise matrix exponentials**
$$
O(d_{\text{hidden}}^2).
$$
All operations are described as **fully parallelizable on GPU** and **linear in sequence length $T$**, with the stated aim of **real-time deployment**. The reported practical performance is **$\sim 1$ s per training iteration** for a **batch of 5 sequences of length 70**, and **$\lesssim 0.1$ s per predicted frame in inference** on a **V100**.

The paper positions OR-GSSM as resolving the **accuracy-real-time trade-off in traditional computational fluid dynamics** for this class of exhaust-vehicle problems. Several distinctions follow directly from the reported construction. First, the method is not presented as a generic black-box temporal predictor, because its hidden dynamics are tied to a state-space representation derived from **semigroup theory** and **Galerkin projection**. Second, it is not equivalent to teacher-forced sequence modeling, because training always uses the model’s own previous prediction. Third, it is not limited to deterministic field prediction, since its validation includes **uncertainty quantification** of flow features and uncertainty distributions. A plausible implication is that OR-GSSM is intended as a surrogate that preserves aspects of numerical and physical structure while reducing the latency associated with high-fidelity CFD.

Source: https://www.emergentmind.com/topics/output-recurrent-gated-state-space-model-or-gssm