T2S-MPC: Neural MPC with Time Embedding

• The paper introduces T2S-MPC, integrating a nominal dynamics model with a time-embedded neural network to predict residual errors.
• It employs a two-timescale update scheme that separates fast adaptation from slower representation learning for rapid response to disturbances.
• Experimental results on a 2D quadrotor show that T2S-MPC reduces tracking error and oscillations compared to baseline MPC methods.

T2S-MPC, short for Time-Embedded Two-Timescale Neural MPC, is an online learning-based model predictive control framework for nonlinear systems with partially known, nonstationary, and potentially unstructured and fast-changing dynamics. It models the discrepancy between a known nominal dynamics model and the true time-varying dynamics as an explicitly time-dependent residual, learns that residual online with a neural network, and integrates the learned model into an MPC loop for repeated receding-horizon optimization. Its defining mechanisms are a sinusoidal time embedding that makes the residual model time-aware and a two-timescale update scheme that separates rapid adaptation from slower representation learning (Shen et al., 24 May 2026).

1. Problem setting and formal control model

T2S-MPC is formulated for systems whose true continuous-time dynamics at time $t$ satisfy

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$

with state $\mathbf{x}(t) \in \mathbb{R}^n$ and input $\mathbf{u}(t) \in \mathbb{R}^m$. The framework assumes the availability of a known nominal model $f(\cdot)$, while the true dynamics are decomposed as

$$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$

where $\mathbf{r}_t$ is an unknown, time-varying residual term representing disturbances, unmodeled physics, or parameter drift (Shen et al., 24 May 2026).

The associated continuous-time optimal control problem minimizes a finite-horizon objective

$$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$

subject to the true dynamics and constraints $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$. Because $f_t$ is both unknown and time-varying, prediction over the horizon is intrinsically difficult. T2S-MPC addresses this by constructing, at each control step, a learned discrete-time model $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$0 and solving a standard receding-horizon MPC problem

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$1

subject to

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$2

Within this formulation, the core novelty is not the MPC objective itself but the construction of $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$3: a nominal model augmented by an online-updated, time-aware residual predictor. This places T2S-MPC within the broader class of residual-learning MPC methods while distinguishing it from approaches that treat model mismatch as stationary or structurally restricted.

2. Time-embedded residual dynamics model

The residual model is trained from online transition tuples of the form

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$4

collected in a dataset

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$5

The paper works with discrete-time residuals, defined for each transition as

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$6

A neural network then approximates this residual through

$$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$7

where $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$8 denotes the network parameters and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$9 is the time embedding (Shen et al., 24 May 2026).

The effective learned dynamics used by MPC are

$\mathbf{x}(t) \in \mathbb{R}^n$0

In a discrete-time state-increment view, this is expressed as

$\mathbf{x}(t) \in \mathbb{R}^n$1

The time embedding is sinusoidal: $\mathbf{x}(t) \in \mathbb{R}^n$2 with

$\mathbf{x}(t) \in \mathbb{R}^n$3

The input to the residual network is the concatenated vector

$\mathbf{x}(t) \in \mathbb{R}^n$4

The intended effect of this construction is explicit time conditioning. Without $\mathbf{x}(t) \in \mathbb{R}^n$5, the network observes only $\mathbf{x}(t) \in \mathbb{R}^n$6, so time variation must be absorbed indirectly through parameter changes induced by online learning. With $\mathbf{x}(t) \in \mathbb{R}^n$7, the residual can be represented as a stationary function in the augmented space $\mathbf{x}(t) \in \mathbb{R}^n$8. The paper states that this provides multi-scale periodic basis functions capable of representing slow drifts, periodic disturbances at various time scales, and superpositions of such effects (Shen et al., 24 May 2026). This suggests that T2S-MPC is designed less around a predefined disturbance model than around a learned temporal basis for nonstationarity.

3. Two-timescale online adaptation

A central design feature of T2S-MPC is its two-timescale update scheme, which partitions the neural parameters as

$\mathbf{x}(t) \in \mathbb{R}^n$9

where $\mathbf{u}(t) \in \mathbb{R}^m$0 are fast parameters and $\mathbf{u}(t) \in \mathbb{R}^m$1 are slow parameters. In the implementation reported in the paper, for a fully connected network with $\mathbf{u}(t) \in \mathbb{R}^m$2 layers, the slow parameters comprise all but the last layer,

$\mathbf{u}(t) \in \mathbb{R}^m$3

while the fast parameters are the final layer,

$\mathbf{u}(t) \in \mathbb{R}^m$4

The forward map uses ReLU activations in the hidden layers,

$\mathbf{u}(t) \in \mathbb{R}^m$5

and a linear output layer,

$\mathbf{u}(t) \in \mathbb{R}^m$6

Training uses a mean-squared residual prediction loss over a batch $\mathbf{u}(t) \in \mathbb{R}^m$7: $\mathbf{u}(t) \in \mathbb{R}^m$8

Fast updates occur every $\mathbf{u}(t) \in \mathbb{R}^m$9 control steps using a sliding window batch $f(\cdot)$0 built from the most recent samples: $f(\cdot)$1 Slow updates occur every $f(\cdot)$2 control steps using a batch $f(\cdot)$3 sampled randomly from the entire buffer: $f(\cdot)$4

The paper’s interpretation is that the final layer acts as a low-dimensional rapid adaptor, while earlier layers preserve a broader and more stable feature representation (Shen et al., 24 May 2026). Using the entire buffer for slow updates is intended to prevent overfitting to the most recent local regime, and updating slow parameters less frequently reduces computational burden. In the reported implementation, optimization uses Adam with $f(\cdot)$5 for fast parameters and $f(\cdot)$6 for slow parameters. There is no explicit forgetting factor; recency weighting is induced implicitly by the use of recent-data batches for fast updates, the larger fast learning rate, and the differing update periods.

A common misunderstanding would be to view this mechanism as equivalent to standard online fine-tuning of a neural dynamics model. The paper explicitly distinguishes T2S-MPC from single-timescale neural MPC by combining explicit time conditioning with parameter splitting. Its claim is not merely faster optimization, but a specific balance between rapid adaptation, stable long-horizon behavior, and lower computational cost.

4. Integration into the MPC loop and reported implementation

At each discrete time $f(\cdot)$7, the T2S-MPC control loop proceeds as follows (Shen et al., 24 May 2026). First, it forms the current dynamics model

$f(\cdot)$8

Second, it solves the discrete MPC problem using horizon $f(\cdot)$9, running cost $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$0, terminal cost $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$1, and constraints $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$2. Third, it applies only the first input $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$3. Fourth, it observes $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$4, appends the new transition to the online dataset, and performs fast and slow parameter updates when their respective update periods are reached.

In the experiments, the control frequency is $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$5, the prediction horizon is $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$6, corresponding to $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$7, and the cost is quadratic: $$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$8 with

$$f_t(\mathbf{x}(t), \mathbf{u}(t)) = f(\mathbf{x}(t), \mathbf{u}(t)) + \mathbf{r}_t(\mathbf{x}(t), \mathbf{u}(t)),$$9

The terminal cost is a quadratic state cost, and the underlying continuous dynamics are integrated via multiple shooting.

The reported network architecture is a fully connected feedforward neural network with $\mathbf{r}_t$0 layers, hidden dimension $\mathbf{r}_t$1, input dimension $\mathbf{r}_t$2, output dimension $\mathbf{r}_t$3, ReLU hidden activations, and linear output. The input dimension corresponds to state $\mathbf{r}_t$4, control $\mathbf{r}_t$5, and time embedding $\mathbf{r}_t$6, giving $\mathbf{r}_t$7. The total parameter count is $\mathbf{r}_t$8. The two-timescale update periods are $\mathbf{r}_t$9 control steps $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$0 and $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$1 control steps $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$2. The source code is stated to be publicly available at the repository linked in the paper abstract (Shen et al., 24 May 2026).

The paper notes that the MPC implementation can handle neural models via automatic differentiation and mentions multiple shooting and L4CasADi in related work, although solver details are not spelled out. A plausible implication is that the framework is intended to be compatible with differentiable NMPC toolchains rather than tied to a bespoke solver design.

5. Experimental setting, baselines, and empirical behavior

Evaluation is conducted in PyBullet on a 2D quadrotor with state

$$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$3

and dynamics

$$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$4

The reported parameters are $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$5 and $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$6, with rotor thrusts $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$7 as inputs (Shen et al., 24 May 2026).

The disturbances are injected as an external acceleration $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$8 on the $$\int_{\tau=t}^{t+t_p} \ell(\mathbf{x}(\tau), \mathbf{u}(\tau))\, d\tau + m(\mathbf{x}(t+t_p))$$9-axis acceleration $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$0. The main disturbance families are linear drift

$$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$1

and periodic disturbance

$$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$2

with additional evaluations on polynomial drift disturbance and linear drift with occasional step changes. The control tasks are stabilization to

$$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$3

circular trajectory tracking in the $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$4 plane, and figure-eight trajectory tracking.

The position error metric is

$$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$5

and the average error over a simulation horizon $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$6 is

$$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$7

Each experiment is repeated for 10 runs with random initial conditions, and results are reported as mean $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$8 standard deviation where applicable.

The baselines are: Nominal MPC, which uses only the nominal model; Neural MPC, which uses a residual network without time embedding and with single-timescale learning; T2S-MPC w/o time embedding, which retains two-timescale updates but removes $$\mathbf{c}(\mathbf{x}(\tau), \mathbf{u}(\tau)) \le 0$$9; T2S-MPC w/o two scales, which retains time embedding but updates all parameters together; and T2S-MPC (full).

The stabilization results under linear and periodic disturbance are summarized below.

Disturbance	Method	Average error
Linear drift	Nominal MPC	$f_t$0
Linear drift	Neural MPC	$f_t$1
Linear drift	T2S w/o time	$f_t$2
Linear drift	T2S w/o two scales	$f_t$3
Linear drift	T2S-MPC	$f_t$4
Periodic	Nominal MPC	$f_t$5
Periodic	Neural MPC	$f_t$6
Periodic	T2S w/o time	$f_t$7
Periodic	T2S w/o two scales	$f_t$8
Periodic	T2S-MPC	$f_t$9

According to the paper, under linear drift both Nominal MPC and Neural MPC show increasing error and oscillations as the dynamics drift, whereas T2S-MPC maintains low, stable error curves. Under periodic disturbances, the baselines exhibit periodic oscillations, and T2S-MPC substantially reduces their amplitude (Shen et al., 24 May 2026). The ablations indicate that both the time embedding and the two-timescale learning mechanism contribute independently, and that the full method performs best when both are combined.

For trajectory tracking, the reported average errors are as follows. Under linear drift, T2S-MPC achieves $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$00 on the circle trajectory and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$01 on the figure-eight, compared with Nominal values of $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$02 and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$03, and Neural values of $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$04 and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$05. Under periodic disturbance, T2S-MPC achieves $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$06 and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$07, compared with Nominal $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$08 and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$09, and Neural $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$10 and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$11. The paper reports tighter tracking and less oscillation in the T2S-MPC trajectories.

A further disturbance ablation varies periodic disturbance amplitude $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$12 and period $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$13, and includes polynomial drift and linear-with-step disturbances. Across all tested cases, T2S-MPC yields lower average error than Nominal and Neural MPC, with especially large gains at higher disturbance magnitudes. The same hyperparameters are used across all conditions, with no retuning, which the paper presents as evidence of robustness and generalization (Shen et al., 24 May 2026).

The reported update-time comparison on an Intel i7-8550U laptop with 8GB RAM gives approximately $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$14 for Neural MPC, $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$15 for the T2S-MPC fast update, and $$\dot{\mathbf{x}}(t) = f_t(\mathbf{x}(t), \mathbf{u}(t)),$$16 for the T2S-MPC slow update. The paper attributes this to the decoupled parameter updates, noting that T2S-MPC has more parameters because of the time embedding but still reduces per-update cost relative to single-scale updates.

6. Interpretation, limitations, and relation to adaptive MPC

T2S-MPC occupies a specific position within learning-based MPC. It is neither a purely nominal controller nor a generic neural dynamics model trained online at a single rate. Its characteristic claim is that explicit time modeling and two-timescale adaptation together improve behavior under unknown, general time-varying dynamics (Shen et al., 24 May 2026). In that sense, it is closer to an adaptive residual-learning MPC architecture than to a conventional robust MPC formulation.

The paper is explicit that its contribution is mainly empirical. It provides no formal stability or regret guarantees, and there are no explicit theorems or lemmas. The design is justified through conceptual arguments—time embeddings improve expressivity for nonstationary residuals, and two-timescale updates resemble classical adaptive control ideas in which slower estimator dynamics support robustness—and through experimental evidence showing stable tracking and regulation across many disturbance regimes without retuning.

This leaves several limitations. First, the method is evaluated on a single platform, a 2D quadrotor in simulation; extension to other systems is presented as natural but is not demonstrated in the paper. Second, although the framework is reported as robust across a broad class of disturbances, the authors explicitly identify improving robustness to high-frequency disturbances as an open problem. Third, the computational burden remains higher than that of pure nominal MPC, even if the decoupled updates are more efficient than full online neural updates. Fourth, formal analysis of convergence for two-timescale learning under nonstationary dynamics and formal stability of MPC with time-embedded residual models are proposed as future directions rather than established results.

A common misconception would be to interpret T2S-MPC as a guaranteed robust adaptive controller. The paper does not make that claim. A more accurate characterization is that it is a practical neural MPC framework for online adaptation under nonstationarity, with performance supported by controlled simulation experiments rather than by formal closed-loop guarantees. Conversely, it would also be inaccurate to reduce it to a simple engineering tweak on neural MPC: the ablation studies are specifically intended to show that the time embedding and the parameter-timescale split are complementary rather than redundant.

Within the reported scope, the method’s conceptual summary is clear: model the true dynamics as nominal plus learned residual, represent time variation through a sinusoidal embedding, and adapt the residual network on two timescales so that the last layer responds quickly to recent changes while earlier layers evolve more slowly from broader experience. The reported outcome is improved stabilization and trajectory tracking under drifting and periodic disturbances, together with robustness across varied disturbance conditions without additional tuning (Shen et al., 24 May 2026).