Synthetic Dynamical Systems
NASCAR with Random Speeds. We evaluate our inference procedure on the NASCAR dataset [ 22 , 28 , 20 ] . Since this system is easily mean-centered, we can isolate the effect of our proposed inference procedure as offset terms are not necessary.
To make this dataset more realistic, we introduce speed variability into the ground truth dynamics as opposed to having a perfect constant speed at all time. Specifically, whenever a trajectory enters a new segment of the track, the system experiences a random change in speed.
We trained all models on 30 trials, each consisting of 1000 time steps (Fig. 2 A) with randomly sampled initial points, and a randomly constructed 10-dimensional linear observation matrix (see Appendix C.1 for full details). Performance is evaluated on 30 held-out trials where the ground truth switching states are defined by the different segments of the track. In all models, we set the M=10 , N=2 and K=4 for DOs or switching states.

Figure 2 B shows that changes in the system speed mask the true transition behavior between segments of the track in rSLDS. Moreover, dLDS identifies coherent segments, but inappropriately learns a different switching pattern for outer and inner edges of the track. In contrast, p-dLDS identifies a switching pattern most consistent with the true track segments despite the presence of noise and randomness in the system’s speed. We note that while there are four true segments, decomposed models form a more parsimonious representation by identifying similar behaviors in different track segments such as in both edges and curves.

Table 1 summarizes our quantitative evaluations on three metrics: 1) the mean squared error (MSE) between the learned and ground truth latent dynamics, 2) the MSE between the inferred and true switch rate to determine agreement of the discrete switching behavior,
and 3) the 100-step inference R^{2} to demonstrate that the learned system generalizes beyond a single step on held-out data. (See Appendix D for mathematical definitions). We see that p-dLDS broadly outperforms existing methods in all metrics and significantly improves inference for decomposed models.

Lorenz System with Random Ramping. Next, we consider the Lorenz system, a chaotic nonlinear system with multiple fixed points, exploring the effect of the offset term. The system is described by the differential equation, \bm{\dot{x}}=\begin{bmatrix}\sigma(x_{2}-x_{1}),x_{1}(\rho-x_{3})-x_{2},x_{1}x_{2}-\beta x_{3}\end{bmatrix}^{\top} where the parameters \rho=28 , \beta=8/3 , and \sigma=10 define a chaotic attractor with two opposing lobes (Fig. 2 D).
We introduce continuous fluctuations in the underlying dynamics by randomly ramping the system’s speed throughout each trajectory. This is accomplished by adjusting the evaluation time intervals given to an ODE solver.
Similar to before, we randomly construct a linear observation function with M=10 and train models on 30 randomly constructed trials with 1000 time points each (see Appendix C.2 ). Furthermore, we define the ground truth switch events as the time points when the signal transitions between the two lobes in addition to the moments when a ramping period concludes.
All models are trained with a latent space of N=3 and K=4 states or DOs.

In figure 2 D, we see that rSLDS does not distinguish between the different speeds along the outer and inner sections of the attractor. Instead, the discrete states obscure the continuum of speeds by incorrectly grouping all activity in each lobe into a single regime.
Furthermore, we observe that dLDS is limited without an offset term, unable to accurately represent multiple fixed points.
Instead of aligning with the two attractor lobes, transitions in the dominant coefficients occur radially relative to the origin and fail to reconstruct the two orbiting fixed points.
Conversely, p-dLDS’s offset term enables learning a system where coefficients better match the true geometry. This representation correctly recovers differences between the outer and inner sections of the attractor while also accurately reconstructing the two orbiting fixed points.
Moreover, this leads to improved estimation of latent dynamics, a switching rate that agrees with the true system, and improved multistep inference performance as shown in Table 1 .