Bilinear Mamba-Koopman Neural MPC for Varying Dynamics

The gist The proposed Bilinear Mamba-Koopman Neural MPC introduces control-dependent coupling in latent dynamics to allow the effective operator to adapt to current inputs, resulting in a model that matches or improves forecasting accuracy across benchmarks and provides robust closed-loop gains under varying conditions.

Bilinear Mamba-Koopman Extension

The authors propose Bilinear Mamba-Koopman Neural MPC as a minimal extension that introduces control-dependent coupling in the latent dynamics, allowing the effective operator to adapt to the current input <ref:2605.04793#pg2> The resulting model is a strict generalization of the standard linear, conditional-independence formulation, adds less than 1% parameters through a low-rank structure, and admits exact model Jacobians that enable efficient Sequential Convex Programming (SCP) with monotonedescent and KKT convergence results under standard trust-region assumptions <ref:2605.04793#pg2>. This extension is defined by the equation zk+1 = diag(A) + Xm i=1 u(i)k Gi z =, Aeff (uk), zk + Bkuk, (1) where A ∈ R dz contains the diagonal continuous-time eigenvalues of the existing Mamba–Koopman backbone (unchanged), Gi ∈ R dz×dz are learned bilinear coupling tensors and Bk ∈ R dz×m is the step-varying actuator matrix from the existing dynamics network <ref:2605.04793#pg2>.

SCP Controller for Bilinear MPC

The model's exact analytical structure enables Sequential Convex Programming (SCP) by exposing exact model Jacobians, which are computed via reverse-mode automatic differentiation of the implemented Lie–Trotter map <ref:2605.04793#pg6> The state Jacobian is denoted as A˜k = ∂FLT / ∂zk (¯zk,u¯k) = EP (¯uk)ED, and the input Jacobian is B˜k = ∂FLT / ∂uk (¯zk,u¯k) <ref:2605.04793#pg6>. The SCP algorithm solves a convex subproblem by eliminating the linearized dynamics to form a dense box-constrained QP solved via OSQP <ref:2605.04793#pg9>. This approach yields monotone descent and KKT convergence results under standard trust-region assumptions, as the predicted QP cost decrease is O(ε 2) uniformly on bounded sets <ref:2605.04793#pg9>.

Empirical Evaluation Across Benchmarks

The empirical evaluation across CartPole and RSCP in time-invariant and time-varying regimes establishes four consistent effects <ref:2605.04793#pg2> These effects include forecasting non-inferiority on every cell when training noise is averaged out, strict gains where control-state coupling is structurally present, training stabilization on RSCP TV, closed-loop MPC wins on RSCP TV when SCP is iterated, and graceful degradation under stale-plan execution <ref:2605.04793#pg2>. Specifically, the bilinear model shows a substantial robustness margin on RSCP TV and a consistent advantage on CartPole TV <ref:2605.04793#pg2>.

Time-Varying Capture and Lead-Time Robustness

The bilinear coupling provides an additional, structured channel through which the operator can adapt to the present operating point in a way that the linear lift cannot reach within a single MPC horizon <ref:2605.04793#pg2>. This is demonstrated in RSCP TV where iterative SCP improves adaptation within the horizon and substantially stabilizes training <ref:2605.04793#pg2>. Furthermore, lead-time commitment unlocks the same correction with a frozen backbone, as the bilinear model’s effective drift Aeff(uk) varies with the committed control uk along the lead window <ref:2605.04793#pg2>. The bilinear model maintains an advantage at every d in the lead-time sweep on RSCP TV, even when d=5, where both models saturate but bilinear still leads <ref:2605.04793#pg9>.

Training Stability and Coupling Strength

The training asymmetry on RSCP TV is explained by the bilinear coupling absorbing time-varying capacity into a stationary operator structure, decoupling the optimization landscape from the data’s non-stationarity <ref:2605.04793#pg2>. The learned ∥G∥F values provide an interpretive diagnostic for which mechanism each cell primarily exercises, such as G serving to absorb operating-point drift under TV physics <ref:2605.04793#pg9> or G absorbing training slack where no physical target exists on RSCP TI <ref:2605.04793#pg9>. The spectral penalty Lstab = X j ReLUλj (Adisc,k) − 1 + ms is used during training to constrain the diagonal entries of A to be at most 1, guaranteeing ρ(Adisc) ≤ 1 <ref:2605.04793#pg9>.

Scope and Limitations

The method inherits the conditional-independence baseline’s assumptions that the latent dynamics are approximately linear at each step and the encoder is expressive enough to find a lifting where this holds <ref:2605.04793#pg2>. The SCP controller provides local (KKT) rather than global optimality guarantees, and convergence rate depends on initialization quality <ref:2605.04793#pg2>. This work positions itself as closing a specific, well-characterized structural gap in the Koopman-for-control literature without claiming a new operator class <ref:2605.04793#pg2>.

Conclusion

The bilinear extension addresses both control–state coupling and timevarying parameters with a minimal architectural change, fewer than 1% added parameters, exact model Jacobians, and a provably convergent SCP controller <ref:2605.04793#pg2>. Across CartPole (which exercises u·x coupling) and RSCP (which exercises time-varying parameters in the TV variant without u·x coupling), four consistent effects emerged <ref:2605.04793#pg2>. These effects include forecasting non-inferiority on all cells under deployment-relevant averaging, substantial training stabilization on RSCP TV, closed-loop MPC gains on RSCP TV when SCP is iterated to convergence, and graceful degradation under stale-plan execution <ref:2605.04793#pg2>. The learned ∥G∥F patterns provide an additional interpretive diagnostic for which mechanism each cell primarily exercises <ref:2605.04793#pg9>. Open directions include extending the bilinear parameterization to the step-varying Bk matrices, imposing structured priors on Gi that encode known physical coupling topology, and designing benchmarks that isolate the u · x structural axis from time-varying parameters more cleanly <ref:2605.04793#pg2>.

AI Assistance Disclosure

We used AI assistants (large language models) for language editing, phrasing, and LaTeX formatting throughout this manuscript <ref:2605.04793#pg2> All technical content, mathematical results, experimental design, code, and analyses are the authors’ own work <ref:2605.04793#pg2>. The AI tools did not contribute novel ideas, proofs, or empirical results <ref:2605.04793#pg2>.

References

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1. Bilinear Mamba-Koopman Neural MPC for Varying Dynamics: The system can adapt to time-varying dynamics within a single MPC horizon by introducing control-dependent coupling in latent dynamics, as shown in equation (1). This allows the effective operator to adapt to the current input, which is structurally impossible under the standard conditional independence constraint.

2. Exact Model Jacobians for Sequential Convex Programming (SCP): The system can solve non-convex MPC problems iteratively via SCP by exploiting exact model Jacobians that enable efficient Sequential Convex Programming (SCP) with monotone-descent and KKT convergence results under standard trust-region assumptions, derived from the bilinear structure in equations (12) to (13).

3. Robust Closed-Loop Performance Under Stale Planning: The system maintains a clear robustness advantage during stale plan execution, as the bilinear model maintains an advantage at every d in lead-time experiments, showing that the coupling provides operator-level correction even with the backbone frozen.

4. Training Stability Enhancement: The system can achieve significantly more reliable training in time-varying regimes by absorbing non-stationarity into a stationary structure, as the bilinear coupling decoupling the optimization landscape from the data’s non-stationarity.