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Deep Bayesian Approaches for Interpretable Nonlinear Structural System Identification

Wei Liu, Konstantinos Vlachas, Eleni Chatzi

arXiv:2610.36633v1cs.LGcs.CEstat.ML

TL;DR

Classical Bayesian filtering relies on prescribed transition and observation models, limiting its fit for identification when governing dynamics are uncertain. This tutorial explains deep Bayesian filters through DMM, PgDMM, and Neural EKF examples, and reports that PgDMM recovers hidden structural states more accurately than Neural EKF in the stated evaluation. The treatment also identifies scalability as a boundary for the low-dimensional formulations considered.

  • Problem

    Classical Bayesian filtering requires prescribed transition and observation models, although identification problems may involve uncertain governing dynamics that cannot be specified a priori.

    Source: §1 · p. 4

  • Method

    The tutorial presents DMM as a data-driven reference and develops PgDMM and Neural EKF as complementary deep Bayesian filtering approaches with physical structure.

    Source: §1 · p. 5

  • Results

    For hidden-state recovery, PgDMM achieves normalized RMSE values of 0.25 for displacement and 0.29 for velocity, compared with 0.41 and 0.34 for Neural EKF, respectively.

    Source: §4.2.3 · p. 25

  • Takeaways & Limitations

    ROM-grounded low-dimensional latent states can provide physical interpretation while learned components accommodate residual dynamics and partially unknown nonlinearities.

    Source: §1 · p. 5

    The formulations treated keep latent dimension small, so larger structures and denser sensor networks call for scalable ROMs and inference schemes.

    Source: §5.3 · p. 35

Abstract

from arXiv · show

System identification, namely the inference of mathematical representations of dynamical systems from measured data, is fundamental to engineering diagnostics, prognostics, and decision-making. Bayesian filtering offers a principled framework for probabilistic state estimation but traditionally relies on prescribed transition and observation models. Recent advances in deep Bayesian filtering address this limitation by embedding learnable representations within probabilistic state-space formulations, enabling data-driven identification of unknown dynamics while retaining physical structure and uncertainty-aware inference. This tutorial provides a systematic introduction to deep Bayesian filtering for nonlinear structural system identification, focusing on the interplay between learned dynamics, physical inductive biases, and posterior inference. The Deep Markov Model serves as a data-driven reference, while the Physics-guided Deep Markov Model (PgDMM) and Neural Extended Kalman Filter (Neural EKF) illustrate complementary approaches based on amortized variational inference and differentiable Kalman filtering, respectively. Their capabilities are examined through dynamic state estimation and joint dynamic state and parameter estimation on a three-dimensional shear-frame benchmark with hysteretic Bouc-Wen connections and sparse acceleration measurements. The results highlight important distinctions between measurement-conditioned reconstruction and open-loop forecasting, demonstrate how physical structure can improve latent-state interpretability without necessarily ensuring predictive fidelity, and reveal differences in the robustness of response and parameter estimation. Practical guidelines for model construction, training, evaluation, and deployment are provided, together with an outlook on remaining challenges in uncertainty calibration, identifiability, and real-world validation.

Glossary of Acronyms

The glossary defines abbreviations used for the paper’s methods, models, and evaluation metrics.

Modeling and learning

  • DL means Deep Learning, and NN means Neural Network.The glossary also expands VAE as Variational AutoEncoder and ROM as Reduced-Order Model.Source: Glossary of Acronyms · p. 1

Filtering methods

  • DMM, PgDMM, and EKF stand for Deep Markov Model, Physics-guided Deep Markov Model, and Extended Kalman Filter.KF is Kalman Filter.Source: Glossary of Acronyms · p. 1

Metrics and other terms

  • MAE and RMSE denote Mean Absolute Error and Root-Mean-Square Error.ELBO is Evidence Lower Bound; DOF is Degree of Freedom; FE is Finite Element; KL is Kullback-Leibler; SHM is Structural Health Monitoring; and SINDy is Sparse Identification of Nonlinear Dynamics.Source: Glossary of Acronyms · p. 1

1 Introduction

The introduction motivates deep Bayesian filtering as a way to learn uncertain dynamics within probabilistic state-space models while retaining physical structure. It presents the DMM, PgDMM, and Neural EKF as representative approaches and frames their structural benchmark as a tutorial demonstration, not an exhaustive survey.

Related approaches

  • The introduction situates the work between fully prescribed mechanistic models and unconstrained black-box learning, which can have limited interpretability or robustness under extrapolation.Hybrid gray-box methods combine prior knowledge with data-inferred components; classical examples include nonlinear subspace identification, NARMAX, wavelet-based methods, and Gaussian processes.Source: §1 · pp. 2–3

Model discovery

  • Symbolic regression and SINDy seek interpretable model forms, but their effectiveness can depend on search spaces, candidate libraries, and feature-selection procedures [57].The tutorial focuses instead on learnable probabilistic observers and their physical or structural biases.Source: §1 · pp. 3, 5

Related limitations

  • Classical Bayesian filters depend on prescribed transition and observation operators, while model-discovery methods may be computationally intensive, data-demanding, or physically uninformative.Physics-informed alternatives prescribe governing equations but may face scalability, generalization, and computational-efficiency challenges.Source: §1 · p. 3

Motivation

  • Deep Bayesian filtering learns parts of the observer from data when adequate transition and observation models cannot be prescribed, while retaining probabilistic uncertainty estimates and possible physical constraints.This addresses nonlinear engineering settings where models must remain meaningful beyond observed responses for tasks such as virtual sensing and decision-making.Source: §1 · pp. 2, 4

Core formulation

  • Deep Bayesian filtering combines learnable observer representations with inference coupled to the probabilistic state-space model, through either a learned variational posterior or differentiable filtering recursions.The paper positions this as a bridge between data-driven schemes and classical Bayesian approaches.Source: §1 · p. 4

Tutorial scope

  • The tutorial develops common principles and practical guidance through a fully data-driven DMM reference and in-depth, complementary PgDMM and Neural EKF examples.The PgDMM uses amortized variational inference, whereas the Neural EKF uses differentiable filtering recursions.Source: §1 · pp. 4–5

Physical structure

  • Physical or structural biases can ground low-dimensional latent states in reduced-order representations while learned components capture residual dynamics and model discrepancy.This construction aims to retain interpretable latent representations while accommodating nonlinearities and partially unknown dynamics.Source: §1 · p. 5

Demonstration

  • The benchmark is a three-dimensional shear frame with hysteretic Bouc-Wen joints under seismic excitation, used to examine latent states, physical parameters, and system responses.The tutorial considers response reconstruction and forecasting, latent-state inference, and parameter estimation rather than exhaustively surveying emerging methods.Source: §1 · p. 5

2 Problem Statement

The problem statement casts identification as inference over unknown dynamics, observations, latent states, and parameters, then formulates learnable probabilistic state-space models and distinct evaluation tasks. It separates fixed-system temporal generalization from joint state-and-parameter estimation.

2.1 System identification as an inverse problem

  • System identification may infer the dynamics, observation mapping, latent states, physical parameters, inputs, or combinations of these, depending on what is known.The deterministic formulation fits candidate transition and observation representations to measurements by minimizing their disagreement.Source: §2.1 · pp. 5–6

2.2 State-space formulation with learnable models

  • The learnable state-space formulation parameterizes transition and observation functions, with process and measurement noise, and can use low-dimensional ROM-based latent states.Its stated assumptions include Gaussian noise, learned covariance matrices, Markov latent states, sparse observations, and a latent dimension small relative to the full-order model.Source: §2.2 · p. 6, Working assumptions · p. 6

2.3 Evaluation protocol

  • The evaluation protocol separates posterior reconstruction using the full measurement window, open-loop forecasting after a measurement-conditioned prefix, and recovery of physical states from latent coordinates.These evaluations can diverge because assimilating measurements may mask transition errors; parameter recovery is added for joint estimation, and each task states its generalization axis.Source: §2.3 · p. 7

2.4 Dynamic state estimation

  • Dynamic state estimation treats system parameters as fixed and evaluates reconstruction, forecasting, and displacement- and velocity-like state recovery from measured input-response sequences.Its protocol assesses temporal generalization for a fixed physical system, not generalization to unseen parameter regimes.Source: §2.4 · p. 7

2.5 Joint dynamic state and parameter estimation

  • Joint dynamic state and parameter estimation recovers time-varying latent dynamics alongside parameters that remain near-constant within each observation window.A parameter latent block represents the physical parameters, and its mapping to those parameters is learned jointly with the framework.Source: §2.5 · p. 7

3 Methodology

The methodology presents deep Bayesian filtering as learnable probabilistic state-space modeling with posterior inference over latent variables. It uses the DMM as a data-driven reference and the PgDMM and Neural EKF as structured alternatives that differ in physical modeling and inference.

Common formulation

  • Deep Bayesian filters share a probabilistic transition model, an observation model, and a mechanism for posterior inference, with formulations differing in parameterization, physical structure, and inference.Transition and observation models are learned from data, while inference may be learned or algorithmic.Source: §3 · pp. 7–8

DMM

  • The DMM serves as a fully data-driven reference for amortized variational inference.The tutorial contrasts it with formulations that incorporate physical knowledge.Source: §3 · p. 8

PgDMM

  • The PgDMM incorporates physical knowledge within a variational deep state-space model.It is developed alongside ROMs as a way to embed low-order physical knowledge into latent dynamics.Source: §3 · p. 8

Neural EKF

  • The Neural EKF learns transition and observation models while using differentiable Kalman filtering recursions for latent-state inference.It is presented as a complementary physics-aware approach to the PgDMM, not as an exhaustive catalog of methods.Source: §3 · p. 8

3.1 Temporal variational state-space skeleton

The section defines a shared probabilistic state-space structure in which latent states evolve from inputs and generate observations, while posterior inference recovers states from observation sequences. DMM, PgDMM, and Neural EKF differ chiefly in how they parameterize the transition, observation, and inference components.

Temporal state-space structure

  • The common skeleton couples a transition density for latent states with an emission density for observations, conditioned on inputs.Under the adopted Gaussian setting, these densities are probabilistic counterparts of the transition and observation functions.Source: §3.1 · p. 8

Posterior inference

  • Posterior inference recovers the latent trajectory from observations and inputs, using an amortized model in DMM and PgDMM or EKF recursions in Neural EKF.The Neural EKF may also use backward smoothing and has no separate inference network.Source: §3.1 · pp. 8–9

Variational learning

  • The shared ELBO rewards observation fidelity while regularizing the inferred posterior toward the transition-model dynamics.Its likelihood term measures reconstruction, and its KL term imposes the dynamics-based regularization.Source: §3.1 · p. 9

Framework distinctions

  • DMM learns all three components without explicit physical bias; PgDMM adds physics-derived structure to transition and observation models, while Neural EKF replaces amortized inference with EKF recursions.The PgDMM retains the DMM’s amortized inference construction, whereas the Neural EKF retains learnable transition and observation models.Source: §3.1 · p. 9

3.2 The Deep Markov Model (DMM)

The DMM is the fully data-driven reference formulation, learning transition, observation, and inference models without prescribing physical meaning for its latent coordinates. Its Gaussian generative models and recurrent amortized inference make the formulation tractable, but the latent representation lacks intrinsic structural correspondence.

DMM formulation

  • The DMM learns its transition, observation, and amortized inference components from data, with no explicit physical bias.It serves as the fully data-driven reference among the formulations considered.Source: §3.2 · p. 9, Figure 1 · p. 10

Generative model

  • Its transition and observation densities are Gaussian, with means and covariances emitted by independent feed-forward neural networks.The formulation assumes Markovian transitions, observations recovered from the current latent state, and time-invariant parameters.Source: §3.2 · p. 9

Inference and training

  • A bidirectional GRU-based recurrent network parameterizes the posterior, and training maximizes the ELBO using reparameterized Monte Carlo samples and a closed-form Gaussian KL term.A combiner fuses recurrent hidden states with the previous latent sample and current control input.Source: §3.2 · p. 9

Variational family and scope

  • Gaussian variational distributions make the KL analytically tractable, while more expressive flow-based families would require Monte Carlo estimation and add computational cost and estimator variance.The paper retains the Gaussian formulation for computational tractability.Source: §3.2 · p. 10

Interpretability boundary

  • DMM latent coordinates have no intrinsic structural correspondence to the underlying physical system.The Neural EKF is introduced as a more structured alternative that can organize latent coordinates as displacement- and velocity-like states.Source: §3.2 · p. 10

3.3 The Neural Extended Kalman Filter (Neural EKF)

The Neural EKF combines learnable transition and observation models with differentiable EKF inference, adding mechanical state structure and replay overshooting for structural response estimation. Its extensions support joint state–parameter estimation, while numerical stability and acceleration-only observability impose practical boundaries.

3.3 The Neural Extended Kalman Filter (Neural EKF)

  • Neural EKF retains learnable transition and observation models but replaces amortized inference with EKF predict-update recursion, optionally followed by backward smoothing.For structural response, it can organize latent coordinates into displacement- and velocity-like components.Source: §3.3 · p. 11

3.3.1 Background: EKF filtering and smoothing

  • The EKF approximates inference by linearizing nonlinear equations and iterating prediction and measurement updates; Rauch-Tung-Striebel smoothing then refines estimates backward over the sequence.Differentiable implementations require safeguards to preserve covariance symmetry and positive semi-definiteness and avoid factorization failures.Source: §3.3.1 · pp. 11–12

3.3.2 Transition and observation models

  • Neural EKF represents transition and observation functions with neural networks trained end-to-end, while its posterior is computed in closed form by the EKF rather than a separate inference network.Process and observation noise are assumed Gaussian with learnable, time-invariant covariances.Source: §3.3.2 · p. 12

3.3.3 Evidence lower bound objective

  • Replay overshooting adds multi-step prediction error because reconstruction-focused variational training can leave the transition model too weak for reliable response prediction.Its contribution must be scheduled carefully: rapid introduction can destabilize optimization, while excessive delay can limit transition learning and long-horizon forecasting.Source: §3.3.3 · p. 13

3.3.4 Dynamic state estimation

  • For dynamic state estimation, the latent state is constrained to an interpretable nodal mechanical state and evolves through a learned acceleration field embedded in a second-order integrator.Acceleration-only observations cannot determine displacement offset and linear drift or velocity offset; the state-recovery comparison removes these components. The semi-implicit Euler scheme also limits accurately represented frequencies; at Δt = 0.01 s, stability requires ωΔt < 2, or approximately 32 Hz.Source: §3.3.4 · pp. 13–14

3.3.5 Joint dynamic state and parameter estimation

  • Joint state and parameter estimation appends a parameter block with identity mean transition and small process noise, propagating parameter uncertainty alongside the dynamic state.A supervised loss on the inferred parameter block is added to the ELBO, enabling joint estimation in one end-to-end procedure.Source: §3.3.5 · p. 15

3.4 Reduced-order modeling for a dynamics-informed latent space

Reduced-order models provide a compact, physically meaningful latent representation for deep Bayesian filtering, connecting reduced states to physical observation channels. For parameter-dependent nonlinear systems, the paper uses interpolated local ROMs and leaves unresolved nonlinearities or model discrepancy to learnable components.

Physics-based reduced-order models

  • A ROM approximates dominant high-dimensional dynamics in a lower-dimensional subspace while retaining connections to physically meaningful states and measurement channels.ROMs may be derived from finite-element models using modal analysis, Proper Orthogonal Decomposition, or Krylov-based reduction.Source: §3.4 · p. 15

Parameter-dependent state-space form

  • ROM operators depend on system parameters, and the discrete observation maps preserve correspondence between their rows and physical degrees of freedom.Different row selections can recover different physical observation channels from the same reduced state.Source: §3.4 · pp. 15–16

Local ROMs in deep Bayesian filtering

  • Because globally accurate ROMs are challenging for nonlinear parameter-dependent systems, the paper adopts interpolated local ROMs and delegates unresolved nonlinearities and model discrepancy to learnable components.This embeds physics-based structure in latent states, transition dynamics, and observation maps without prescribing the entire model.Source: §3.4 · p. 16

3.5 The Physics-guided Deep Markov Model (PgDMM)

The PgDMM combines physics-guided and learnable components to produce a physically interpretable latent representation while modeling discrepancies and uncertainty. It applies this framework to dynamic state estimation and joint state-and-parameter estimation.

3.5 The Physics-guided Deep Markov Model (PgDMM)

  • The PgDMM combines physical knowledge in the generative model with learnable components for dynamics and discrepancies not captured by the prescribed physics.Its physical component can take different forms; the applications use a reduced-order model.Source: §3.5 · p. 16

3.5.1 Transition model

  • Its transition combines physics-guided and neural random-variable contributions, with trainable mixing and variance components.The physics branch anchors the latent state, while the neural component captures discrepancy; diagonal covariance is used under the stated white-noise, low-noise-to-signal assumption.Source: §3.5.1 · pp. 16–17

3.5.2 Observation model

  • The observation model maps latent states to measurements through a parameterized distribution, with a learnable head absorbing discrepancy from the ROM readout.For the structural-response setting, the emission has an additive physics-derived component and a learned discrepancy component.Source: §3.5.2 · p. 17

3.5.3 Inference model

  • The approximate posterior uses a bidirectional recurrent network to process the observation sequence, with shared recurrent parameters across its physics and neural inference streams.The streams differ through their weights and biases despite using the same bidirectional network.Source: §3.5.3 · p. 18

3.5.4 Evidence lower bound objective

  • The PgDMM maximizes an ELBO that balances expected log-likelihood reconstruction against KL regularization, including a physical-constraint term.The objective follows maximum-likelihood estimation with a variational posterior approximation because the true latent posterior is intractable.Source: §3.5.4 · p. 18

3.5.5 Dynamic state estimation

  • Dynamic state estimation uses a single-branch, balanced reduced mechanical state and an amortized posterior; acceleration measurements alone constrain the unobserved displacement and velocity.The latent state comprises 16 relative modal displacements and 16 modal velocities; its acceleration emission is constrained by fixed stiffness and the measurements.Source: §3.5.5 · p. 19
  • Random rollout origins and progressively increasing horizons yielded one of the most substantial forecasting improvements in the experiments.The passage notes that rollout-origin and horizon choices can matter more for forecasting accuracy than the weighting coefficient λfc.Source: §3.5.5 · p. 20

3.5.6 Joint dynamic state and parameter estimation

  • Joint estimation augments the dynamic latent state with a parameter block that evolves near-identity, encoding the assumption that target parameters remain constant over the observation window.At inference, the trained inference model recovers the parameter block from the response signal alone.Source: §3.5.6 · p. 20

4 Demonstration on a Benchmark Shear Frame With Hysteretic Links

The demonstration uses a three-dimensional two-story, two-bay shear-frame model with nonlinear Bouc-Wen joints. Its simulated responses vary with structural and excitation parameters and are observed through sparse acceleration sensors.

4 Demonstration on a Benchmark Shear Frame With Hysteretic Links

  • The benchmark is a three-dimensional, two-story, two-bay shear frame with nonlinear joints exhibiting Bouc-Wen hysteresis.The model is presented as a demonstrative nonlinear structural-dynamics case study.Source: §4 · p. 20

4.1 Model description

  • Each joint’s restoring force combines linear and nonlinear terms and depends on nodal displacement and a hysteretic deformation component.The shape, smoothness, and amplitude of the hysteretic curve depend on β, γ, m, and A.Source: §4.1 · p. 21
  • The frame uses two parallel x-axis frames spaced 5 m apart, with two 7.5 m bays per frame and 3.2 m stories, under base excitation at π/4 to the longitudinal axis.The figure schematic marks sensor locations with blue crosses.Source: §4.1 · p. 21, Figure 2 · p. 21
  • The simulated system varies Young’s modulus and Bouc-Wen stiffness, while five sensors record longitudinal and transverse accelerations with additive white noise.The passage specifies temperature-dependent modulus variation, sampled stiffness weighting, and channel noise with standard deviation equal to 5% of each channel’s RMS.Source: §4.1 · pp. 21–22

4.2 Dynamic state estimation

The dynamic state estimation experiments compare measurement-conditioned reconstruction, open-loop forecasting, and recovery of unobserved structural states. Reconstruction is accurate, but forecasting and state recovery expose important differences between the frameworks and limits of physical structure.

4.2 Dynamic state estimation

  • The task fixes physical parameters; posterior reconstruction uses the complete observation window, whereas open-loop forecasting uses only the warm-up prefix and receives no later measurement correction.Structural state recovery maps inferred displacement–velocity states to physical coordinates; both quantities are unobserved and are detrended before scoring.Source: §4.2 · p. 22

4.2.1 Setup

  • Training and testing use chronological windows from the same physical system, not unseen parameter values, and the frameworks receive unequal prior physics that favors PgDMM’s free-response behavior.PgDMM receives fixed modal stiffness, damping, and base forcing and uses ten symplectic substeps; Neural EKF identifies its linear backbone and uses one step per interval.Source: §4.2.1 · p. 22, Table 2 · p. 22

4.2.2 Reconstruction and open-loop forecasting

  • On the held-out window, reconstruction normalized RMSE is 0.020 (PgDMM) and 0.039 (Neural EKF), while 1.8-s forecast RMSE is 0.102 and 0.109, respectively.Both reconstruction errors are below the 0.042 measurement-noise standard deviation, while open-loop errors are approximately three to five times larger.Source: §4.2.2 · pp. 22–23, Table 3 · p. 23

4.2.3 Recovery of the hidden structural state

  • For detrended hidden-state recovery, PgDMM normalized RMSE is 0.25 for displacement and 0.29 for velocity, versus Neural EKF values of 0.41 and 0.34.PgDMM displacement errors range from 0.20 to 0.31 across channels, compared with 0.18 to 0.58 for Neural EKF.Source: Table 3 · p. 23, §4.2.3 · p. 25

4.2.2 Reconstruction and open-loop forecasting

  • Neural EKF has lower open-loop forecast RMSE at 0.2 s (0.049 versus 0.068) and 1.0 s (0.066 versus 0.079), but the frameworks are nearly tied over 1.8 s.PgDMM is less accurate longitudinally (0.119 versus 0.081 transverse), while Neural EKF is less accurate transversely (0.124 versus 0.091 longitudinal); both lose longitudinal amplitude while keeping response phase.Source: §4.2.2 · p. 23
  • Prescribed linear operators can slow forecast-error growth without preserving physical fidelity: PgDMM’s learned residual yields 1–3% damping ratios in its three lowest modes, versus 0.1–0.2% in the reduced model.The resulting excessive damping makes the predicted response decay too rapidly.Source: §4.2.2 · p. 24

4.2.3 Recovery of the hidden structural state

  • PgDMM’s prescribed stiffness constrains poorly observable displacement components, while Neural EKF’s slow modes from 0.01 to 1.3 Hz permit drift below the 3.9-Hz first natural frequency.The acceleration-only record leaves a linear displacement trend and constant velocity undetermined; PgDMM detrending changes its reported errors by less than 10%.Source: §4.2.3 · pp. 24–25

4.2.4 Training behavior

  • PgDMM validation forecast error declines to 0.10 at epoch 14,700, while Neural EKF reaches 0.081 at iteration 5,500 and then oscillates between 0.089 and 0.102.Each validation and test curve uses one window, and their full-suffix rankings reverse: Neural EKF leads validation (0.081 versus 0.103), while PgDMM leads test (0.102 versus 0.109).Source: §4.2.4 · p. 26

4.3 Joint dynamic state and parameter estimation

The joint-estimation experiments compare posterior response reconstruction and physical-parameter recovery across three frameworks on held-out test data. Typical reconstruction errors are similar, but their upper tails differ, while parameter accuracy varies by parameter and framework.

4.3.1 Setup

  • The test set comprised 29 held-out runs, partitioned by run from 190 simulations; each run supplied ten acceleration channels and three system parameters.The experiment separately evaluates posterior reconstruction and parameter recovery; open-loop forecasting is not repeated because parameter estimates remain fixed after measurement updates cease.Source: §4.3 · p. 26, §4.3.1 · p. 27, Table 4 · p. 27

4.3.2 Response reconstruction

  • The three methods’ reconstructed trajectories closely followed reference responses, while DMM slightly underestimated the largest acceleration peaks.Figure 7 shows two representative test sequences; the quantitative evaluation uses 2 s windows.Source: §4.3.2 · p. 27, Figure 7 · p. 28
  • The 90th-percentile normalized RMSE was 0.055 for DMM, 0.038 for PgDMM, and 0.025 for Neural EKF; 10.8%, 7.9%, and 0.5% of windows, respectively, exceeded 0.05.Median normalized RMSEs were between 0.011 and 0.012, so the upper tails distinguish reliability more clearly than typical-window performance.Source: §4.3.2 · p. 28, Figure 8 · p. 29
  • The largest channel-wise errors occurred at the third and fifth monitored nodes for DMM and PgDMM, and at the first monitored node for Neural EKF.The distributions were broadly comparable across channels rather than dominated by one measurement location.Source: §4.3.2 · pp. 28–29, Figure 9 · p. 29

4.3.3 Parameter estimation

  • All nine model–parameter combinations had MAE below 1.0% of the parameter range, and the estimates closely followed reference values across the sampled ranges.The parity plots and residuals cover all testing windows.Source: §4.3.3 · p. 30, Figure 11 · p. 30
  • Neural EKF improved excitation-amplitude error by about an order of magnitude over DMM and PgDMM, but had the largest mean error and heaviest upper tail for the Bouc-Wen stiffness factor.Young’s modulus weighting-factor recovery was comparable across frameworks; PgDMM’s excitation-amplitude error was 1.6 × 10−3.Source: §4.3.3 · pp. 30–31

4.4 Interpretability of the inferred latent space

The latent-space analyses show that PgDMM’s prescribed modal organization supports a more direct physical interpretation than the unstructured learned representations. This organization is evident in concentrated variance and frequency patterns, but spectral alignment alone does not establish unique physical-mode identification.

4.4 Interpretability of the inferred latent space

  • In a representative joint-estimation test run, the first three PgDMM modal displacement coordinates accounted for approximately 99% of latent displacement variance.The passage cautions that latent-coordinate variance is not physical modal energy.Source: §4.4 · p. 31, Figure 12 · p. 31
  • In dynamic state estimation, the first three PgDMM modal coordinates matched reduced-model modes at 3.9, 5.2, and 6.3 Hz and jointly accounted for 58% of modal-displacement-coordinate variance.Eight additional coordinates near 11.5 Hz contributed 27%; the remaining five predominantly followed excitation content around 1 Hz.Source: §4.4 · p. 32, Figure 13 · p. 32
  • PgDMM latent spectra were predominantly narrow-band around lower ROM resonances, whereas DMM and Neural EKF spread frequency content across coordinates without an evident modal ordering.The compared architectures achieved comparable reconstruction accuracy on the considered realization, and spectral alignment does not prove unique identification of physical modes.Source: §4.4 · pp. 32–33, Figure 14 · p. 33

5 Discussion

The discussion argues that model structure, inference mechanisms, and training objectives determine what deep Bayesian filters can represent and deliver. It recommends evaluating models for their intended use and identifies uncertainty calibration, scalability, changing systems, and validation beyond simulation as unresolved boundaries.

5.1 Key insights

  • Prescribed physical structure narrows latent-state ambiguity and can make coordinates physically readable, but learned corrections can alter prescribed dynamics and should be checked.The discussion recommends imposing physical knowledge where its validity is established and verifying the influence of learned components.Source: §5.1 · pp. 33–34
  • Amortized inference supports flexible, scalable posterior approximation but may generalize poorly or produce noncausal smoothing, while Neural EKF filtering is causal but relies on Gaussian local linearization and costly covariance computations.The suitable inference mechanism depends on application, latent dimension, and temporal processing requirements.Source: §5.1 · p. 34
  • Measurement-conditioned reconstruction can remain accurate despite unreliable open-loop forecasts, so predictive objectives should match the deployment horizon and parameter objectives must support identifiability.Supervised latent-parameter estimation requires labeled configurations; marginal-likelihood filtering can avoid direct parameter labels when observations identify the parameters.Source: §5.1 · p. 34
  • Latent interpretability requires physically tied coordinates, observable directions, and verification against states that were not measured.The discussion warns that acceleration-only data leave displacement ambiguous up to an offset and linear trend, and learned transitions may introduce weakly observed drifting modes.Source: §5.1 · p. 34

5.2 Practical guidelines for training and deployment

  • Training and deployment guidance includes matching discretization and scales to prescribed physics, selecting objectives and checkpoints for intended use, and inspecting learned dynamics and parameter identifiability.Evaluation should distinguish reconstruction, forecasting, state recovery, and parameter recovery across multiple realizations and relative to measurement noise.Source: §5.2 · p. 35

5.3 Outlook

  • The tutorial’s uncertainty distributions are not calibrated in the presented work and omit epistemic uncertainty from learned networks.Calibration and comparison with sensor characteristics remain important before uncertainty estimates inform monitoring or operational decisions.Source: §5.3 · p. 35
  • The demonstrations use simulated data from one benchmark structure, so they illustrate capabilities and limitations rather than establish a general ranking.The discussion calls for field measurements and shared benchmarks; current formulations also assume fixed systems within observation windows and measured inputs, while larger structures challenge small-latent and cubic-cost approaches.Source: §5.3 · pp. 35–36

CRediT authorship contribution statement

The CRediT statement assigns conceptualization, methodology, software, formal analysis, investigation, visualization, and writing contributions across the authors, with data curation assigned to Konstantinos Vlachas and supervision and funding acquisition to Eleni Chatzi.

CRediT authorship contribution statement

  • Wei Liu, Konstantinos Vlachas, and Eleni Chatzi share conceptualization, methodology, investigation, and writing-review contributions, with additional roles distributed across software, analysis, visualization, data curation, supervision, and funding.The statement assigns software and visualization to Liu and Vlachas, formal analysis to Liu and Chatzi, data curation to Vlachas, and supervision and funding acquisition to Chatzi.Source: CRediT authorship contribution statement · p. 36

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