Federated Nonlinear System Identification

• Federated nonlinear system identification is a framework for collaboratively estimating nonlinear dynamics from distributed clients while preserving data privacy.
• It employs two regimes: one using direct least-squares with known nonlinear feature maps, and another using a learned Koopman surrogate with Bayesian filtering.
• The method leverages local FedAvg or deep network updates to achieve faster convergence and enhanced sample efficiency across diverse client datasets.

Federated nonlinear system identification studies how to estimate nonlinear dynamical models from multiple distributed clients without transmitting raw trajectories or observations. In the formulations currently documented on arXiv, the problem appears in at least two technically distinct regimes. One regime assumes a shared family of linearly parameterized nonlinear dynamics with known real-analytic features and client-specific parameters, so that federated learning can be analyzed relative to centralized least squares (Tupe et al., 20 Aug 2025). A second regime treats partially observed nonlinear systems and learns a common Koopman-space linear surrogate by combining state estimation via the unscented Kalman filter and the unscented Rauch-Tung-Striebel smoother with federated training of a deep Koopman network (Chen et al., 7 Jul 2025). Across both regimes, the core objective is collaborative identification with privacy preservation, while the main technical differences concern state availability, model parameterization, excitation assumptions, and the form of the convergence guarantees.

1. Problem formulations

In the linearly parameterized setting, there are $M$ clients indexed by $i=1,\dots,M$, and each client evolves according to

$$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$

Here $x_t^{(i)}\in\mathbb{R}^{n_x}$ is the state, $u_t^{(i)}\in\mathbb{R}^{n_u}$ the control input, $w_t^{(i)}$ the process noise, $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$ a known, real-analytic feature map, and $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$ unknown client-specific parameters. The formulation also imposes bounded heterogeneity through

$$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$

Each client collects $N_i$ independent trajectories of length $i=1,\dots,M$0 under i.i.d. inputs and locally solves a least-squares problem

$i=1,\dots,M$1

while the global objective is to find a single parameter matrix $i=1,\dots,M$2 minimizing aggregate prediction error across all clients; in centralized form this is $i=1,\dots,M$3 (Tupe et al., 20 Aug 2025).

In the partially observed Koopman setting, the unknown nonlinear system is

$i=1,\dots,M$4

where $i=1,\dots,M$5 is unobserved, $i=1,\dots,M$6 is known, and $i=1,\dots,M$7 is a partial observation. Process noise and observation noise are Gaussian, independent across time and clients, and each client may use a different observation map $i=1,\dots,M$8 and different noise covariances. The federated goal is to learn a common linear surrogate in lifted space,

$i=1,\dots,M$9

without sharing raw observation data $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$0 or estimated states $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$1 (Chen et al., 7 Jul 2025).

2. Architectural variants

The two formulations instantiate different architectural choices for federated nonlinear identification. The first is a direct parameter-estimation pipeline built around least squares on a fixed nonlinear feature map. The second is a representation-learning pipeline in which nonlinear dynamics are linearized in a learned Koopman space and the latent-state training data are obtained by Bayesian filtering and smoothing. Taken together, these works delineate two complementary operating regimes: direct federated estimation when a suitable feature map is known, and federated surrogate learning when only observations are available and the state must first be reconstructed.

Dimension	Linearly parameterized formulation	KF-FedKL formulation
State availability	$$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$2 is used directly	only $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$3 is observed
Model form	$$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$4	$$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$5
Local estimation object	least-squares estimate $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$6	local DKN parameter $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$7
Shared structure	common feature map $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$8	common lifted linear surrogate
Raw-data sharing	not transmitted	not transmitted

The distinction is not merely implementation-level. In the first case, nonlinear system identification is reduced to linear estimation in a fixed nonlinear basis, so the theory centers on excitation and Gram-matrix invertibility. In the second case, the identification target is itself a learned lifting, so the theory combines stochastic optimization, client selection, data heterogeneity, and filtering error (Tupe et al., 20 Aug 2025, Chen et al., 7 Jul 2025).

3. Federated optimization mechanisms

For linearly parameterized nonlinear systems, the federated algorithm adopts a standard server-client FedAvg style framework specialized to least squares. The server initializes $$x_{t+1}^{(i)}=\theta^{(i)*}\,\phi\bigl(x_t^{(i)},u_t^{(i)}\bigr)+w_t^{(i)}.$$9 and a learning rate $x_t^{(i)}\in\mathbb{R}^{n_x}$0, broadcasts $x_t^{(i)}\in\mathbb{R}^{n_x}$1 at each round, and every client sets $x_t^{(i)}\in\mathbb{R}^{n_x}$2. Client $x_t^{(i)}\in\mathbb{R}^{n_x}$3 then performs $x_t^{(i)}\in\mathbb{R}^{n_x}$4 local gradient-descent steps,

$x_t^{(i)}\in\mathbb{R}^{n_x}$5

returns $x_t^{(i)}\in\mathbb{R}^{n_x}$6 to the server, and the server aggregates according to

$x_t^{(i)}\in\mathbb{R}^{n_x}$7

In compressed notation, the local batch-gradient update is $x_t^{(i)}\in\mathbb{R}^{n_x}$8 with $x_t^{(i)}\in\mathbb{R}^{n_x}$9 and $u_t^{(i)}\in\mathbb{R}^{n_u}$0 (Tupe et al., 20 Aug 2025).

For Kalman Filter aided Federated Koopman Learning, each client $u_t^{(i)}\in\mathbb{R}^{n_u}$1 in round $u_t^{(i)}\in\mathbb{R}^{n_u}$2 collects $u_t^{(i)}\in\mathbb{R}^{n_u}$3 new observations with success probability $u_t^{(i)}\in\mathbb{R}^{n_u}$4, runs UKF+RTS to obtain $u_t^{(i)}\in\mathbb{R}^{n_u}$5 estimated states, adds them to a local buffer $u_t^{(i)}\in\mathbb{R}^{n_u}$6, trains its local deep Koopman network for $u_t^{(i)}\in\mathbb{R}^{n_u}$7 epochs using SGD or Adam with batch size $u_t^{(i)}\in\mathbb{R}^{n_u}$8 and learning rate $u_t^{(i)}\in\mathbb{R}^{n_u}$9, clears the local buffer, and sends the updated model back to the server. The server activates a subset $w_t^{(i)}$0 by a policy $w_t^{(i)}$1, such as a threshold on $w_t^{(i)}$2, random selection, or round-robin, and aggregates by

$w_t^{(i)}$3

Only model updates leave the clients, which the formulation identifies explicitly as the privacy mechanism (Chen et al., 7 Jul 2025).

4. Convergence analysis and finite-sample guarantees

The finite-sample theory for linearly parameterized nonlinear dynamics is organized around five assumptions: $w_t^{(i)}$4 is real-analytic and bounded; process noise is iid, sub-Gaussian with variance bounded below by $w_t^{(i)}$5; inputs are iid with variance bounded below by $w_t^{(i)}$6; local dynamics are LISS; and parameter heterogeneity is at most $w_t^{(i)}$7. Under open-loop random inputs, Lemma 1 establishes a block-martingale small-ball condition: there exist constants $w_t^{(i)}$8 and $w_t^{(i)}$9 such that, for every client $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$0 and any unit vector $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$1,

$$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$2

This implies a $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$3-BMSB condition for each regressor sequence. Proposition 1 then gives a sufficient sample condition,

$$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$4

under which, with probability at least $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$5, every local design matrix satisfies

$$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$6

and the pooled Gram matrix obeys $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$7, with $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$8. Proposition 2 bounds the noise-regressor cross-term, and Theorem 1 yields, with probability at least $$\phi:\mathbb{R}^{n_x}\times\mathbb{R}^{n_u}\to\mathbb{R}^{n_\phi}$$9, an error bound consisting of a variance term

$\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$0

and a heterogeneity term $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$1, where $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$2 and $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$3. The paper further states that the convergence rates in the linear and nonlinear cases differ only by a constant, and that this constant depends on the feature map $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$4. When $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$5 and $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$6 is fixed, the first term scales as $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$7, which is described as a $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$8 speedup in sample efficiency over any single client (Tupe et al., 20 Aug 2025).

The KF-FedKL analysis uses a different abstraction. Under assumptions that each local loss is $\theta^{(i)*}\in\mathbb{R}^{n_x\times n_\phi}$9-smooth and $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$0-strongly convex, stochastic gradients have bounded variance $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$1, and the data error induced by the Kalman filter enters as zero-mean gradient noise with variance $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$2, the “virtual” global parameter sequence $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$3 satisfies

$$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$4

Here $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$5 captures data heterogeneity and $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$6 depends on the selection probability $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$7. The stated conclusion is linear convergence up to a neighborhood proportional to gradient and observation noise (Chen et al., 7 Jul 2025).

5. Feature design, lifting, and state recovery

In federated nonlinear least squares, the feature map $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$8 is central because it enters the rate constants through $$\max_{i,j}\|\theta^{(i)*}-\theta^{(j)*}\|\le \epsilon.$$9, $N_i$0, and $N_i$1. The constants $N_i$2 and $N_i$3 quantify how exciting the regressors are in every direction, while $N_i$4 is an upper bound on $N_i$5 and controls the heterogeneity-bias constant $N_i$6. The documented design guidance is explicit: real-analytic feature maps guarantee the BMSB property under i.i.d. inputs; examples include monomials $N_i$7 and trigonometric functions such as $N_i$8 and $N_i$9. Polynomial bases of moderate degree capture smooth nonlinearities at the cost of larger $i=1,\dots,M$00, whereas trigonometric or RKHS-inspired maps can offer controlled norms while preserving excitation. In practice, $i=1,\dots,M$01 is chosen to match physical knowledge, such as including $i=1,\dots,M$02 for a pendulum, and to maximize $i=1,\dots,M$03 while keeping $i=1,\dots,M$04 moderate (Tupe et al., 20 Aug 2025).

In the Koopman-based formulation, the analogous design object is the lifted representation learned by a deep autoencoder. An encoder $i=1,\dots,M$05 and decoder $i=1,\dots,M$06 define $i=1,\dots,M$07, and learning minimizes a composite loss

$i=1,\dots,M$08

where $i=1,\dots,M$09 penalizes lifted linear-dynamics error, $i=1,\dots,M$10 reconstruction error, and $i=1,\dots,M$11 end-to-end prediction error. Because clients observe only $i=1,\dots,M$12, state reconstruction precedes learning: the unscented Kalman filter propagates sigma points through $i=1,\dots,M$13 and $i=1,\dots,M$14 to obtain filtered estimates $i=1,\dots,M$15, and the unscented RTS smoother optionally runs backward over a window of length $i=1,\dots,M$16 to produce smoothed states $i=1,\dots,M$17. These smoothed states replace the unknown $i=1,\dots,M$18 in the training loss. The reported ablation result that all three loss terms $i=1,\dots,M$19 are needed for a physically meaningful Koopman representation underscores that representation quality and identification quality are coupled in this setting (Chen et al., 7 Jul 2025).

6. Empirical behavior, operating regimes, and limitations

The reported experimental evidence for linearly parameterized nonlinear federated identification covers a synthetic 3-state system with sine nonlinearity, an inverted pendulum with 1 state, and a quadrotor with 13 states. In all cases, inputs have the form $i=1,\dots,M$20, where $i=1,\dots,M$21 is a nominal control policy and $i=1,\dots,M$22 is an i.i.d. zero-mean disturbance, a setting described as ensuring “non-active exploration.” Trajectories of length $i=1,\dots,M$23 or $i=1,\dots,M$24 are generated, each client collects $i=1,\dots,M$25 such trajectories under perturbed parameters $i=1,\dots,M$26 with $i=1,\dots,M$27, and feature maps include real-analytic terms such as $i=1,\dots,M$28 and $i=1,\dots,M$29 for the pendulum and Euler-angle sines and cosines for the quadrotor. The empirical findings are consistent across several axes: with fixed per-client data, the normalized estimation error

$i=1,\dots,M$30

decays approximately proportional to $i=1,\dots,M$31; at fixed $i=1,\dots,M$32, increasing $i=1,\dots,M$33 reduces error in the classical $i=1,\dots,M$34 fashion; larger $i=1,\dots,M$35 increases the bias floor $i=1,\dots,M$36; a small number of local gradient-descent steps, approximately $i=1,\dots,M$37–$i=1,\dots,M$38, reduces communication while preserving convergence; and too many local steps can induce client drift under heterogeneity. The experiments also report that a single client, $i=1,\dots,M$39, has much slower error decay, whereas even modest client counts, approximately $i=1,\dots,M$40–$i=1,\dots,M$41, yield large reductions in sample complexity per client (Tupe et al., 20 Aug 2025).

The KF-FedKL experiments evaluate Van der Pol oscillator, single pendulum, Lorenz63, YNI cardiac cell, and double pendulum systems, with the double pendulum using real video data. The baselines are single-client, perfect-data, and centralized training. The reported metrics are one-step held-out test loss $i=1,\dots,M$42, a multi-step prediction error $i=1,\dots,M$43 combining state-space and lifted-space terms, and communication cost approximated by one model upload per active client per round. The main findings are that federated KF-FedKL nearly matches centralized performance in test loss and multi-step error, single-client training dramatically underperforms, and the effect of Kalman-filter estimation error is mild because “perfect data” is only marginally better. Among UKF, UKF with resampling, EKF, and Particle Filter, the vanilla UKF is reported to strike the best accuracy/variance trade-off at moderate noise. Parameter sweeps further show that more data and higher success probability help up to overfitting, heterogeneity slows convergence, and more local epochs accelerate local fitting at the cost of communication (Chen et al., 7 Jul 2025).

A recurrent misconception is that federated nonlinear identification improves monotonically with arbitrary increases in local computation or client diversity. The reported results do not support that simplification. In the least-squares formulation, benefits saturate when heterogeneity $i=1,\dots,M$44 is large, and excessive local steps produce client drift. In the Koopman formulation, heterogeneity enters the convergence neighborhood through $i=1,\dots,M$45, while partial observations introduce an additional noise contribution through the filtering stage. This suggests that the practical regime of highest benefit is one in which clients are sufficiently aligned to permit averaging, excitation is preserved through feature or lifting design, and state uncertainty or client-selection noise remains controlled.