SIVA: Physics-Guided System Identification

• The paper introduces SIVA, a framework that couples parameter estimation, model validation, and uncertainty quantification using a physics-guided adversarial approach for structural dynamics.
• The methodology employs a neural network-based parameter generator and a discriminator that validates synthetic accelerations produced by known equations of motion, ensuring generalizable updates.
• Empirical case studies on Duffing oscillators and cantilever beams demonstrate SIVA’s high accuracy in parameter updating and robust uncertainty quantification across unseen validation conditions.

System Identification via Validation and Adaptation (SIVA) is a physics-guided framework for learning dynamical-model parameters directly from data while coupling three tasks that classical pipelines often separate: parameter estimation, model validation on independent data, and uncertainty quantification. In its structural-dynamics formulation, a neural network maps random noise to physically meaningful parameters, a known equation of motion converts those parameters into synthetic accelerations, a mean square error term enforces agreement with training data, and a discriminator operating on unseen validation data guides the parameter generator toward parameter sets that generalize beyond the calibration condition (López et al., 25 Jun 2025). A closely related formulation treats SIVA as a model-updating method for nonlinear structures with an existing reduced-order physics model, especially when a nominal linear finite-element description has been altered by added mass or nonlinear attachments (López et al., 30 Jul 2025).

1. Definition and conceptual scope

SIVA was introduced in structural system identification as a parametric, physics-guided, adversarial method for identifying the governing parameters of a structural dynamics model directly from measured response data (López et al., 25 Jun 2025). The method is motivated by the observation that structural identification usually involves partially known physics rather than a wholly unknown input–output map. For this reason, SIVA does not learn an arbitrary black-box predictor of accelerations; it learns parameters that remain interpretable within a known equation of motion.

The structural papers place SIVA against the standard taxonomy of parametric, nonparametric, and semi-parametric identification. Parametric methods assume that the mathematical model is known and estimate coefficients; nonparametric methods infer dynamics directly from data; semi-parametric methods combine both. SIVA is designed for the regime in which the governing model, or at least a reduced-order form of it, is already available, but key coefficients are uncertain or must be updated after structural modification (López et al., 30 Jul 2025). In the cantilever-beam application, the starting point is an already identified linear FE model reduced by methods such as Guyan reduction or SEREP, and SIVA updates it to represent a lumped mass and nonlinear end attachment (López et al., 30 Jul 2025).

A recurring misconception is to treat SIVA as an ordinary GAN applied to time series. The structural papers are explicit that the generated objects are not images or text, but physically meaningful parameters that drive the equation of motion; the discriminator evaluates the realism of the resulting accelerations, not the plausibility of parameters in isolation (López et al., 30 Jul 2025). The method is therefore adversarial only at the validation layer; its identification mechanism remains anchored in the governing dynamics.

2. Mathematical structure and identification mechanism

In the structural formulation, the parameter-generator network $P$ maps a latent Gaussian vector to unknown physical coefficients,

$$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$

with $\mathbf{z}$ generated in Python with randn and seed 42 in the nonlinear beam study (López et al., 30 Jul 2025). The network reported in the structural papers is a fully connected multilayer perceptron with layers $64 \to 32 \to 16 \to n$, LeakyReLU hidden activations with slope $0.2$, and a linear output layer; for parameters spanning many orders of magnitude, scientific-notation parameterizations such as $k=a\cdot 10^b$ are used (López et al., 25 Jun 2025).

The generated parameters are inserted into a known equation of motion to produce synthetic accelerations. In the general structural formulation, the multiple-degree-of-freedom acceleration is written as

$$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$

where $\mathbf{q}$ is displacement, $\dot{\mathbf{q}}$ velocity, $\mathbf{M}$ the known mass matrix, $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$0 the damping matrix, and $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$1 the stiffness matrix (López et al., 25 Jun 2025). In the cantilever-beam model-updating variant, the reduced-order dynamics are written as

$$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$2

with $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$3, $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$4, and $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$5 inherited from the known linear reduced model, and $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$6 containing the unknown nonlinear attachment terms (López et al., 30 Jul 2025).

The discriminator $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$7 receives acceleration signals and outputs a scalar probability through a fully connected architecture $$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$8 with LeakyReLU hidden activations and a sigmoid output (López et al., 30 Jul 2025). Its loss is binary cross-entropy,

$$\boldsymbol{\theta} = P(\mathbf{z}), \qquad \mathbf{z}\sim\mathcal N(0,1),$$9

and the beam paper states that the optimal discriminator loss under indistinguishability is $\mathbf{z}$0 (López et al., 30 Jul 2025). The parameter-generator loss combines an adversarial term and an MSE term,

$\mathbf{z}$1

with $\mathbf{z}$2 in the beam study (López et al., 30 Jul 2025). The structural SIVA paper presents the same logic as

$\mathbf{z}$3

thereby making identification explicitly dependent on both training-data fidelity and adversarial validation (López et al., 25 Jun 2025).

3. Validation, adaptation, and uncertainty quantification

The defining feature of SIVA is that validation is built into parameter learning rather than deferred to a separate post hoc check. Independent validation datasets are passed through the physics model with generated parameters, and the resulting accelerations are judged by the discriminator as real or fake (López et al., 25 Jun 2025). In the cantilever-beam formulation, the discriminator sees real training accelerations, model-generated accelerations from training data, and model-generated accelerations from validation data; the generator is therefore pressured not only to fit the calibration signal but also to produce parameters whose induced responses remain plausible under unseen excitation conditions (López et al., 30 Jul 2025).

This validation mechanism is the source of the “adaptation” component in the method’s name. The parameter generator is adapted by feedback from validation data rather than by training-only error minimization. In structural model updating, this is especially relevant when a nominal model is already available but attachments, nonlinearities, or added masses have altered the actual system (López et al., 30 Jul 2025). The method is not merely selecting coefficients that minimize simulation mismatch on one trajectory; it is adapting the coefficient distribution toward generalization across independent datasets.

Uncertainty quantification follows from the stochastic parameter generator. After convergence, the papers describe two primary procedures. In Approach I, the trained generator is sampled 1000 times and the mean of the parameter samples is used. In Approach II, training is continued after convergence, parameter values from later epochs are recorded, and those values are averaged. For UQ, a normal distribution is fit to each parameter sample set using MATLAB’s fitdist, the PDF is evaluated with pdf over $\mathbf{z}$4 standard deviations, the mean is marked by a dashed red line, a 95% confidence interval is shown in green, and the exact value is marked by a black line (López et al., 30 Jul 2025). The structural SIVA paper adds a third practical selection rule, Approach III, which chooses the candidate parameter set minimizing displacement MSE after full time integration (López et al., 25 Jun 2025).

A further point emphasized by the Duffing study is robustness to overtraining. Training losses fluctuate heavily at first, as expected in an adversarial setup, but converge after roughly 400 epochs, and training can continue to epoch 2000 with only slight changes in the mean parameter values (López et al., 25 Jun 2025). This stability is one reason the authors regard post-convergence sampling as meaningful for UQ.

4. Structural case studies and reported results

The empirical basis for SIVA is concentrated in nonlinear structural dynamics, where the method has been tested on analytical and experimental systems and on nonlinear model updating (López et al., 25 Jun 2025).

System	Identification and validation setup	Reported outcome
Duffing oscillator	$\mathbf{z}$5 s simulation at $\mathbf{z}$6 kHz; validation with different initial conditions; convergence after roughly $\mathbf{z}$7 epochs	Identified parameters very close to ground truth; validation trajectories match exact responses; stable behavior up to epoch $\mathbf{z}$8
Two-degree-of-freedom coupled oscillators	Free-response training; validation on $\mathbf{z}$9 N and $64 \to 32 \to 16 \to n$0 N impacts; convergence after about $64 \to 32 \to 16 \to n$1 epochs	Good reproduction of unseen responses; stronger advantage under the $64 \to 32 \to 16 \to n$2 N, more nonlinear case
Nonlinear cantilever beam	Training with a $64 \to 32 \to 16 \to n$3 kN half-sine impact; validation with $64 \to 32 \to 16 \to n$4 kN and $64 \to 32 \to 16 \to n$5 kN impacts	Accurate parameter estimation and model updating; close exact/identified responses in time and frequency domains; SINDy performs poorly in this case

For the analytical Duffing oscillator,

$64 \to 32 \to 16 \to n$6

the reported parameters are $64 \to 32 \to 16 \to n$7 kg, $64 \to 32 \to 16 \to n$8 Ns/m, $64 \to 32 \to 16 \to n$9 Ns/m, $0.2$0 N/m, and $0.2$1 N/m (López et al., 25 Jun 2025). Validation uses initial conditions different from training. The paper reports parameter estimates such as $0.2$2 to $0.2$3 Ns/m, $0.2$4 to $0.2$5 Ns/m, $0.2$6 to $0.2$7 N/m, and $0.2$8 to $0.2$9 N/m in the best SIVA variants (López et al., 25 Jun 2025).

For the experimental coupled-oscillator system, the identified free-response model is validated under hammer impacts of $k=a\cdot 10^b$0 N and $k=a\cdot 10^b$1 N that were not used in identification (López et al., 25 Jun 2025). The paper reports that both methods align well in the weakly nonlinear $k=a\cdot 10^b$2 N case, whereas under the $k=a\cdot 10^b$3 N impact, where nonlinearity is stronger, SIVA captures the linear-oscillator dynamics better, particularly in wavelet-domain features (López et al., 25 Jun 2025).

The nonlinear cantilever-beam study extends the same logic to model updating. The beam is uniform, homogeneous steel, modeled with 15 Euler–Bernoulli elements, reduced to translational DOFs, and augmented at its free end by a linear spring $k=a\cdot 10^b$4, a cubic spring $k=a\cdot 10^b$5, and a lumped mass $k=a\cdot 10^b$6 (López et al., 30 Jul 2025). Training uses a half-sine impulsive force of amplitude $k=a\cdot 10^b$7 kN and duration $k=a\cdot 10^b$8 s; the transient is simulated in MATLAB with ode45, tolerances $k=a\cdot 10^b$9, total time $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$0 s, and sampling rate $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$1 kHz, while validation uses $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$2 kN and $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$3 kN impacts (López et al., 30 Jul 2025). The paper reports close agreement between exact and identified tip displacements, matching Fourier spectra, strong validation performance under unseen impacts, and a comparison in which SINDy struggles on the highly nonlinear attachment problem (López et al., 30 Jul 2025).

5. Broader identify–validate–adapt literature

Beyond the named structural framework, several papers instantiate a SIVA-style coupling of identification, validation, and adaptation, but they do so with different mathematical objects and validation criteria. In-context system identification with encoder–decoder Transformers treats learning as a meta-learning problem over a distribution $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$4 of system datasets and then adapts the pretrained meta-model to a specific system, a shifted class, or a new prediction horizon. There, validation enters through early stopping on a separate validation dataset, with reported minima after roughly $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$5 iterations for an in-class Wiener–Hammerstein case and around $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$6 iterations for an out-of-class Parallel Wiener–Hammerstein case; task-change adaptation from $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$7-step to $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$8-step prediction turns an otherwise difficult training problem into a feasible one, moving from average RMSE about $$\ddot{\mathbf{q}}=\mathbf{M}^{-1}\left[-\mathbf{B}\dot{\mathbf{q}}-\mathbf{K}\mathbf{q}+\mathbf{F}\big(t,(\cdot)\big)\right],$$9 to about $\mathbf{q}$0 on the harder long-horizon task (Piga et al., 2023).

A related transfer-learning line for recurrent neural networks starts from a validated nominal RNN and augments it after regime change with an additive correction term learned by Jacobian Feature Regression. The correction is fit by regularized least squares in Jacobian feature space, admits an RNTK-GP interpretation, and yields evaluation $\mathbf{q}$1 above $\mathbf{q}$2 on both outputs of the perturbed CSTR benchmark; on a nonlinear RLC circuit, sensitivity-based Jacobian computation is about $\mathbf{q}$3 faster than naive construction, while full retraining can require about $\mathbf{q}$4 larger runtime for only slight additional accuracy in some cases (Forgione et al., 2022).

Other SIVA-style formulations shift the validation criterion itself. Identification through invariant measures replaces pointwise trajectory matching with comparison between observed occupation measures and simulated stationary measures from a Fokker–Planck or Perron–Frobenius surrogate, then adapts the discretization through data-adaptive unstructured meshes and the coordinates through Takens delay embeddings to improve scalability and identifiability (Botvinick-Greenhouse, 31 Jan 2025). LILAD jointly learns a dynamics model and a Lyapunov certificate through in-context learning, adapts both by a short prompt rather than gradient updates, and validates stability through a test-time attenuator $\mathbf{q}$5 that enforces a sufficient Lyapunov decrease condition even under out-of-task variation (Jena et al., 26 Nov 2025). In structural input–state estimation, Universal Filtering and Universal Smoothing validate estimator behavior experimentally on a five-storey shear frame and introduce a self-tuning mechanism that adapts the process covariance online instead of relying on offline tuning (Liu et al., 20 Aug 2025).

These works indicate that “validation and adaptation” is not tied to a single diagnostic. Depending on the application, validation may take the form of adversarial discrimination, early stopping, invariant-measure discrepancy, Lyapunov decrease, residual whiteness, or experimental cross-checking on real hardware.

6. Limitations, misconceptions, and open directions

The principal limitation stated by the structural SIVA papers is dependence on a sufficiently complete governing model. SIVA assumes that the equation of motion is known; its performance degrades if essential physics are missing from the model (López et al., 30 Jul 2025). The beam study is explicit that noise handling is not yet robust, that experimental noise and insufficient smoothing can make the method struggle, and that the use of a standard GAN leaves training instability as an open concern (López et al., 30 Jul 2025). Computational cost is another practical issue: the same paper notes that SINDy is much faster computationally, producing results in a fraction of a second, whereas SIVA Approach II took about $\mathbf{q}$6 hr $\mathbf{q}$7 min for $\mathbf{q}$8 epochs, although SIVA simultaneously provides validation and UQ (López et al., 30 Jul 2025).

Another misconception is to treat SIVA as synonymous with any adversarial simulator-identification method. SimGAN, for example, identifies a hybrid physics simulator by adversarial reinforcement learning and then refines a control policy in the learned simulator without interleaving further target-data collection, but its validation signal is a discriminator-derived trajectory realism reward rather than the structural parameter-validation mechanism used in SIVA (Jiang et al., 2021). Conversely, some practical identification loops fit the SIVA idea without using adversarial learning at all. The gas power plant study based on GGOV1 and ST6B performs least-squares pre-estimation, Cuckoo Search parameter adaptation, validation on a separate dataset, and residual whiteness analysis at confidence level $\mathbf{q}$9, explicitly forming an iterative identification–validation–adaptation loop on Chabahar power plant field data (Hosseinalizadeh et al., 2018).

A broader open direction concerns automation of the loop itself. ASIA delegates hypothesis formation, code modification, and validation-driven iteration to an LLM-based coding agent, but its authors note implicit test leakage, reduced methodological transparency, and reproducibility concerns (Piga et al., 11 May 2026). This suggests that SIVA can be read in two ways: narrowly, as a specific physics-informed adversarial framework for structural dynamics, and more broadly, as a research design principle in which identification is repeatedly checked against evidence outside the fitting signal and then adapted accordingly.