OR-GSSM: Physics-Informed Recurrent SSM

• The paper introduces OR-GSSM, a physics-informed recurrent model that leverages state-space formulations to capture multiphase flows in exhaust vehicles.
• It combines the gas-liquid Navier-Stokes equations with a Gated State Space Transition unit and output recursion to ensure numerical consistency and stable training.
• Empirical validation shows OR-GSSM achieves higher accuracy and faster runtimes compared to OR-ConvLSTM and OR-ConvGRU in modeling underwater and water-exit scenarios.

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 (Chen et al., 1 Aug 2025).

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 $x_t\in\mathbb{R}^{n_x}$0-theorem. The output is

$x_t\in\mathbb{R}^{n_x}$1

namely the dimensionless physical fields—velocity components, pressure, volume fraction—on the Eulerian grid (Chen et al., 1 Aug 2025).

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 $x_t\in\mathbb{R}^{n_x}$2, the GSST takes the new external input $x_t\in\mathbb{R}^{n_x}$3 (identified in the detailed exposition with $x_t\in\mathbb{R}^{n_x}$4) and the previous hidden state $x_t\in\mathbb{R}^{n_x}$5 and computes

$x_t\in\mathbb{R}^{n_x}$6

$x_t\in\mathbb{R}^{n_x}$7

$x_t\in\mathbb{R}^{n_x}$8

$x_t\in\mathbb{R}^{n_x}$9

$u_t\in\mathbb{R}^{n_u}$0

$u_t\in\mathbb{R}^{n_u}$1

Here, $u_t\in\mathbb{R}^{n_u}$2 denotes a spatial convolution, $u_t\in\mathbb{R}^{n_u}$3 is the element-wise sigmoid, $u_t\in\mathbb{R}^{n_u}$4 ensures $u_t\in\mathbb{R}^{n_u}$5, the tensors $u_t\in\mathbb{R}^{n_u}$6 are trainable convolution kernels, the functions $u_t\in\mathbb{R}^{n_u}$7 are pointwise nonlinearities such as Swish, and $u_t\in\mathbb{R}^{n_u}$8 is the hidden dimension.

The paper assigns specific state-space interpretations to these quantities. By design, $u_t\in\mathbb{R}^{n_u}$9 plays the role of $y_t\in\mathbb{R}^{n_y}$0 in a zero-order-hold discretization of the continuous SSM, while $y_t\in\mathbb{R}^{n_y}$1 when $y_t\in\mathbb{R}^{n_y}$2 is small. The gate $y_t\in\mathbb{R}^{n_y}$3, described as inspired by GRU, adaptively mixes old and new information (Chen et al., 1 Aug 2025).

This architecture is therefore not merely a gated RNN cell with renamed variables. The explicit parameterization of $y_t\in\mathbb{R}^{n_y}$4, $y_t\in\mathbb{R}^{n_y}$5, and $y_t\in\mathbb{R}^{n_y}$6 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

$y_t\in\mathbb{R}^{n_y}$7

$y_t\in\mathbb{R}^{n_y}$8

$y_t\in\mathbb{R}^{n_y}$9

where $w_t$0 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 $w_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 $w_t$2 acquires extra terms from the recurrence path, and that these terms dynamically constrain parameter updates and mitigate long-term error accumulation (Chen et al., 1 Aug 2025).

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 $w_t$3 regularization. The regularization coefficient is reported as

$w_t$4

Optimization uses Adam with an initial learning rate of $w_t$5, halved every 500 epochs, for a total of 2 500 epochs. The batch size is 5 sequences. The reported architecture uses hidden dimension $w_t$6, 4 transposed-conv layers in $w_t$7, and a spatial grid of $w_t$8 (Chen et al., 1 Aug 2025).

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 $w_t$9, recycling $v_t$0 at each step.

The training and inference loops given in the detailed exposition are operationally simple: initialize the parameters

$v_t$1

set the initial hidden state to zero, encode the input, propagate with GSST, form the masked output $v_t$2, accumulate the sequence loss, and update $v_t$3 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 $v_t$4 mm, $v_t$5 mm, and exhaust seam $v_t$6 mm at 19 mm is simulated using high-fidelity CFD (VOF + $v_t$7–$v_t$8 SST). Metrics are reported over 600 test samples as time-averaged $v_t$9-norm relative error in percent, together with per-iteration runtime on an NVIDIA V100.

Model	Error profile $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$0	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 (Chen et al., 1 Aug 2025).

For the underwater cone-head validation, the computational domain is $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$1 m on a $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$2 grid with velocity inlet, pressure outlet, symmetry axis, and no-slip walls. Training uses 100 CFD runs over

$x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$3

Testing and uncertainty quantification use 600 runs at

$x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$4

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 $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$5 m, $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$6 m, seam at 2.5 m, internal cavity $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$7, and $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$8 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 $x_t := \{U(t),\alpha(t)\}\in\mathbb{R}^{2N},$9–$U$0 SST for hydrodynamics. Training uses 100 runs over

$U$1

while testing and uncertainty quantification use 200 runs at

$U$2

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 $U$3 confidence bands. In the water-exit hemisphere-head case, statistics at fixed depths

$U$4

show larger uncertainty near the separating interface and backflow region (Chen et al., 1 Aug 2025).

The computational profile given in the paper identifies the per-step cost as dominated by convolutional blocks

$U$5

and channel-wise matrix exponentials

$U$6

All operations are described as fully parallelizable on GPU and linear in sequence length $U$7, with the stated aim of real-time deployment. The reported practical performance is $U$8 s per training iteration for a batch of 5 sequences of length 70, and $U$9 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.