Neural Extended Kalman Filters for Learning and Predicting Dynamics of Structural Systems

Variational autoencoders (VAEs) kingma2013auto are popularly used for capturing useful latent representations z from data x by reconstructing the VAE input to an output vector. The VAE can be extended to a dynamical version by considering the temporal evolution of the latent variables girin2020dynamical . Among various dynamical VAEs, Deep Markov Models (DMMs krishnan2017structured ) offer a primary example of unsupervised training of a deep state-space model. By chaining an approximate inference model with a generative model and using the variational lower bound maximization approach, DMMs deliver the most direct extension of static VAEs to account for temporal data girin2020dynamical . The DMM couples a transition model $p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1})$, describing temporal evolution of the latent states, with an observation model $p_{\theta}(\mathbf{z}_{t}|\mathbf{x}_{t})$, serving as a decoder and describing the process from latent states to observations, and an inference model $q_{\phi}(\mathbf{z}_{t}|\mathbf{x}_{1:T})$, serving as an encoder, which approximates the true posterior distribution $p(\mathbf{z}_{t}|\mathbf{x}_{1:T})$.

The training of a VAE consists in maximizing a variational lower bound of the data log-likelihood $\log p(\mathbf{x})$. Upon application of the variational principle on the inference model $q_{\phi}(\mathbf{z}_{t}|\mathbf{x}_{1:T})$, the evidence lower bound (ELBO) of the data log-likelihood is given as follows:

where the KL-divergence is defined as $\text{KL}[q(\mathbf{z})||p(\mathbf{z})]:=\int q(\mathbf{z})\log\frac{q(\mathbf{z})}{p(\mathbf{z})}d\mathbf{z}$.

Apart from the parameters $\theta$ involved in transition and observation models, the VAE architecture typically employs an additional inference network, which is independently parameterized by the hyper-parameters $\phi$ of the corresponding neural networks. The key idea of VAE is to approximate the posterior distribution with the introduced inference network. However, the objective function ELBO largely depends on the quality of the inference model through the first reconstruction term, which does not involve a transition model, and the influence of the transition models is indirectly relegated to a regularizer for the inference model, reflected in the second KL-divergence term li2021replay . The KL-divergence itself is a regularization term and it attempts to only make the single-step dynamics transition similar to the posterior given by the inference network. The above arguments indicate that the quality of VAE reconstructions, as measured by the value of the objective ELBO, strongly depends on the inference model but rather weakly on the transition models. This characteristic of the training objective makes the learned model capable of delivering accurate inference results under availability of observation (monitoring) data, but unsuitable for prediction due to the insufficient training of the transition models.

As shown in Fig. 3, the result demonstrates the advantages of the Neural EKF over the DMM, which is, as mentioned, a dynamical VAE model that directly extends the static VAE to account for temporal data. The results are demonstrated on a 2-DOF Duffing system, defined by Equation (28) appearing in the Applications section. It is observed that the DMM fails to deliver sufficiently accurate predictions, while the predicted responses, furnished by the Neural EKF, very well fit the true (reference) responses.

For nonlinear state estimation and parameter identification in civil and mechanical engineering, the Extended Kalman Filter (EKF) has been a popular choice for weakly nonlinear systems, mainly due to its ease of implementation, robustness and suitability for real-time applications. It assumes a sequence of measurements $\mathbf{x}_{1:T}$ from a monitored dynamical system, which are generated by some latent states $\mathbf{z}_{1:T}$ that are not necessarily directly observed. The transition from a state $\mathbf{z}_{t-1}$ to the next state $\mathbf{z}_{t}$ is termed as transition, and the process from a state ${z}_{t1}$ to its corresponding observation $\mathbf{x}_{t}$ is termed as observation. Compared to dynamical VAEs, the EKF assumes that the transition and observation models are given, as described by the following two equations:

where $f$ and $g$ are two known linear/nonlinear functions governing the transition and observation models, and $w_{t}\sim\mathcal{N}(0,\mathbf{Q}_{t})$, $v_{t}\sim\mathcal{N}(0,\mathbf{R}_{t})$ are Gaussian noise sources with mean zero and covariances $\mathbf{Q}_{t}$ and $\mathbf{R}_{t}$, respectively. EKF assumes that the inferred posterior distributions of latent states are based on past and current observations $\mathbf{x}_{1:t}$ and driving force $\textbf{u}_{1:t}$, and follow a Gaussian distribution:

Here $\bm{\mu}_{t|t}$ and $\mathbf{\Sigma}_{t|t}$ represent the mean and covariance of the posterior distribution. The Kalman Filter (KF) 10.1115/1.3662552 offers a closed-form of the posterior distributions $q(\mathbf{z}_{t}|\mathbf{x}_{1:t},\mathbf{u}_{1:t})$ of latent states and provides exact inference for linear systems. The EKF conducts approximate inference, employing a linearization of the nonlinear equations. For EKF, the inference of the posterior distribution is obtained by iteratively executing the following two steps. The first step predict is to compute the posterior distribution $q(\mathbf{z}_{t}|\mathbf{x}_{1:t-1},\mathbf{u}_{1:t-1})=\mathcal{N}(\mu_{t|t-1},\mathbf{\Sigma}_{t|t-1})$, which is simply based on previous observations $\mathbf{x}_{1:t-1}$:

where $\mathbf{A}_{t-1}=\frac{\partial f(\bm{\mu}_{t-1|t-1},\mathbf{u}_{t-1})}{\partial\bm{\mu}_{t-1|t-1}}$ is the Jacobian of $f$ at $\bm{\mu}_{t-1|t-1}$. The second step update pertains in updating the posterior distribution from the predict step using the current observation $\mathbf{x}_{t}$:

where $\mathbf{C}_{t}=\frac{\partial g(\bm{\mu}_{t|t-1})}{\partial\bm{\mu}_{t|t-1}}$ is the Jacobian of $g$ at $\bm{\mu}_{t|t-1}$. The Jacobian matrices $\mathbf{A}_{t}$ and $\mathbf{C}_{t}$ are used as linear approximations of the original nonlinear functions $f$ and $g$, respectively, for conveniently computing the involved covariances $\mathbf{\Sigma}_{t|t-1}$ and $\mathbf{\Sigma}_{t|t}$.

The EKF is a recursive filtering method for conducting inference based on $\mathbf{x}_{1:t}$, i.e., the observations until the present time $t$. Since we assume that a sequence of observations $\mathbf{x}_{1:T}$ is available, it is possible and reasonable to further update the inference of posterior distributions with the whole sequence of observations, with respect to a time step within the sequence. This process is termed as smoothing. The Rauch-Tung-Striebel smoothing rauch1965maximum is a Kalman smoothing algorithm to infer such posterior distributions $q(\mathbf{z}_{t}|\mathbf{x}_{1:T},\mathbf{u}_{1:T})=\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$ of latent states based on the whole sequence of available observations. The smoothing process can fully make use of available data when computing the posterior distributions and generate more accurate inference results with adequate information, instead of only utilizing information before each time step, which is the case for the Kalman filtering process. Note that, compared to Eq.(4), now the posterior depends on the whole sequence of $\textbf{x}_{1:T}$ and $\textbf{u}_{1:T}$. Once the EKF process is completed, the smoothing algorithm operates in a backward manner, from time $T$ to $1$, to update the posteriors:

The intermediate quantities $\bm{\mu}_{t|t}$, $\bm{\mu}_{t+1|t}$, $\mathbf{\Sigma}_{t|t}$ and $\mathbf{\Sigma}_{t+1|t}$ in this smoothing process are already available at Eqs.(5)-(9), in the EKF stage.

Here we introduce the prior and posterior distributions that will be used for computing the loss function of Neural EKFs in the later section. The final posterior distribution after application of the EKF algorithm and the smoothing step is given as $q(\mathbf{z}_{t}|\mathbf{x}_{1:T},\mathbf{u}_{1:T})=\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$, where $\bm{\mu}_{t|T}$ and $\mathbf{\Sigma}_{t|T}$ are given in Eqs.(11) and (12). Furthermore, given $\mathbf{z}_{t-1}\sim\mathcal{N}(\bm{\mu}_{t-1|T},\mathbf{\Sigma}_{t-1|T})$, we can compute the distribution of each evolved $\textbf{z}_{t}$:

where $\mathbf{A}_{t-1|T}=\frac{\partial f(\bm{\mu}_{t-1|T},\mathbf{u}_{t-1})}{\partial\bm{\mu}_{t-1|T}}$ is the Jacobin of $f$ at $\bm{\mu}_{t-1|T}$. Subsequently, the distribution of observation $p(\mathbf{x}_{t}|\mathbf{z}_{t})$, given $\mathbf{z}_{t}\sim\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$, can be derived as

where $\mathbf{C}_{t|T}=\frac{\partial f(\bm{\mu}_{t|T},\mathbf{u}_{t})}{\partial\bm{\mu}_{t|T}}$.

In Eqs.(2) and (3), if we replace $f$ and $g$ by learnable transition and observation functions, the framework can be described as:

where $f_{\theta_{\textbf{t}}}$ and $g_{\theta_{\textbf{o}}}$ are the learnable functions governing the transition and observation models, both parameterized by neural networks with parameters $\theta=\theta_{\textbf{t}}\bigcup\theta_{\textbf{o}}$. The process noise sources $w_{t}$ and observation noise $v_{t}$ are assumed to follow Gaussian distributions, with respective time-invariant covariances, i.e., $w_{t}\sim\mathcal{N}(0,\mathbf{Q})$ and $v_{t}\sim\mathcal{N}(0,\mathbf{R})$ for all time steps $t$. The time-invariant covariances $\mathbf{Q}$ and $\mathbf{R}$ are also set as learnable parameters during the training process. It is noted that here the latent dynamics is formulated as a discrete model. Alternatively, continuous models such as Neural Ordinary Differential Equations chen2018neural can also be implemented to model the latent dynamics if one wants to treat the latent dynamics as a continuous model, as explained in Appendix.

The inference model of the Neural EKF follows the format of the EKF. This is different from the inference model in VAEs, where $q_{\phi}$ is a separate inference network parameterized by $\phi$ independent of parameters within $f_{\theta_{t}}$ and $g_{\theta_{o}}$. Since the objective ELBO largely depends on the goodness of reconstruction and inference, a separate inference network is thus weakening the training of the transition and observation models.

With Kalman filters conducting inference, the parameters to be learned are summarized in the vector $\theta$, which includes neural network parameters of both the transition model $f_{\theta_{\mathbf{t}}}$ and the observation model $g_{\theta_{\mathbf{o}}}$. In addition, the initial values $\bm{\mu}_{0|0}$ and $\mathbf{\Sigma}_{0|0}$ and covariances $\mathbf{Q}$, $\mathbf{R}$ for respective noises are also parameters to be learned. Similar to the VAE, the training of Neural EKFs is embedded in the variational inference methodology, with the EKF algorithm charged with conducting inference. Given any inference model $q_{\phi}(\mathbf{z}_{t}|\mathbf{x})$ and Markovian property implied by the dynamics, the ELBO in Eq.(1) can be expressed in a factorized form ($\mathbf{x}_{1:T}$ and $\mathbf{u}_{1:T}$ are abbreviated as $\mathbf{x}$ and $\mathbf{u}$ in the following formulations for simplicity):

where, in the Neural EKF, $q_{\phi}(\mathbf{z}_{t}|\mathbf{x},\mathbf{u})$ is actually $q_{\theta}(\mathbf{z}_{t}|\mathbf{x},\mathbf{u})$, which alleviates the requirement of additional parameters, further to $\theta$, within the transition and observation models, and thus $\mathcal{L}(\theta,\phi;\mathbf{x})$ reduces to $\mathcal{L}(\theta;\mathbf{x})$. Since the posterior distributions $q_{\theta}(\mathbf{z}_{t}|\mathbf{x},\mathbf{u})$ can be computed in closed form by EKF, the distributions in Eq.(17) can thus be computed explicitly given $\mathbf{z}_{t}\sim\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$, as shown in Section Prior and posterior distributions. Thus, the ELBO Eq.(17) can be computed in a surrogate way as:

The above involved distributions are based on the prior $\mathbf{z}_{t}\sim\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$. The log-probability and KL-divergence can be computed analytically when all the involved distributions are Gaussian. The log-probability of a Gaussian distribution $p(\mathbf{x})=\mathcal{N}(\bm{\mu},\mathbf{\Sigma})$ has an explicit format as:

where $d_{x}$ is the dimension of $\mathbf{x}$. As stated in Eq.(14), since $\mathbf{z}_{t}\sim q_{\theta}(\mathbf{z}_{t}|\mathbf{x},\mathbf{u})=\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$ and replace $g$ by $g_{\theta_{\textbf{o}}}$, we have $p(\mathbf{x}_{t}|\mathbf{z}_{t})=\mathcal{N}(g_{\theta_{\textbf{o}}}(\bm{\mu}_{t|T}),\mathbf{C}_{t|T}\Sigma_{t|T}\mathbf{C}_{t|T}^{T}+\mathbf{R})$ . Using Eq.(19), the log-likelihood term $\log p_{\theta_{\textbf{o}}}(\mathbf{x}_{t}|\mathbf{z}_{t})$ in Eq.(18) can be computed approximately as:

Similarly, the KL-divergence term $\text{KL}(q(\mathbf{z})||p(\mathbf{z}))$, when $p(\mathbf{z})=\mathcal{N}(\bm{\mu}_{p},\mathbf{\Sigma}_{p})$ and $q(\mathbf{z})=\mathcal{N}(\bm{\mu}_{q},\mathbf{\Sigma}_{q})$ are both Gaussian distributions, comprises an analytical form as:

where $d_{z}$ is the dimension of $\mathbf{z}$. As stated in Eq.(13), and replace $f$ by $f_{\theta_{\textbf{t}}}$,

since we have $\mathbf{z}_{t-1}\sim q_{\theta}(\mathbf{z}_{t-1}|\mathbf{x},\mathbf{u})=\mathcal{N}(\bm{\mu}_{t-1|T},\mathbf{\Sigma}_{t-1|T})$ and $\mathbf{z}_{t}\sim q_{\theta}(\mathbf{z}_{t}|\mathbf{x},\mathbf{u})=\mathcal{N}(\bm{\mu}_{t|T},\mathbf{\Sigma}_{t|T})$. Substitute these mean and covariance into Eq.(21), the KL-divergence term in Eq.(18) can be computed as:

Thus the final objective ELBO enforced by EKF and smoothing formulas can be written as:

Typically, the variational objective function for the dynamical VAE framework focuses on the reconstruction loss, which largely depends on the inference model (encoder) and observation model (decoder), whereas the transition model plays a minor role in the variational objective and works as an intermediate process for training. This often results in an accurate inference model and a less meaningful transition model, which is not applicable for prediction because the transition model cannot reflect the true underlying dynamics. Therefore, it is necessary that the objective function also takes the accuracy of the transition model into consideration to ensure its closeness to the true latent dynamic process. To address this issue, we adopt the overshooting method proposed in li2021replay , which is termed as replay overshooting. The key point lies in simply introducing the prediction loss of the generative model into the objective function. After obtaining a sequence of means and covariances $\{(\mu_{t|T},\Sigma_{t|T})\}$, the initial value $(\bar{\mu}_{0},\bar{\Sigma}_{0})=(\mu_{0|T},\Sigma_{0|T})$ will be further used for prediction process. Then similarly as the prediction step in EKF, the predicted values are obtained via the generative model as:

and we receive another set of distributions $\bar{q}(\mathbf{z}_{t})=\mathcal{N}(\bar{\mu}_{t},\bar{\Sigma}_{t})$ from the generative model. The final reconstruction loss is composed of the both sets of posterior distributions, i.e., the posterior distributions obtained from the Kalman inference model in Eq.(20), as well as those obtained from the transition model, weighted by $\alpha$ (in this paper, we set $\alpha=0.5$). These form the objective function combined with the KL loss, expressed as:

where the first and third term are the same as in (24), while the second term is for overshooting.

The pipeline of the Neural EKF is summarized in Algorithm 1.

Once a Neural EKF model is trained, the learned function $f_{\theta_{\textbf{t}}}$ and $g_{\theta_{\textbf{o}}}$ is available and can be used for response prediction. The prediction is performed using $\mathbf{z}_{t}=f_{\theta_{\textbf{t}}}(\mathbf{z}_{t-1},\mathbf{u}_{t-1})$ and $\mathbf{x}_{t}=g_{\theta_{\textbf{o}}}(\mathbf{z}_{t})$, given a new initial condition $\textbf{z}_{0}$ and corresponding driving forces $\textbf{u}_{1:T}$. Note that a new initial condition can be either specified by the user, or inferred from actual data via use of the inference model of the Neural EKF.

The Neural EKF is capable of generating predictions in real-time to detect structural anomalies and changes as they occur for structural health monitoring. On the other hand, it requires offline training on a characteristic batch of operational data, prior to it being applied in prediction mode. However, in intervals where loads significantly increase with respect to the training set, possibly triggering more severe nonlinearity, or if the system experiences damage, the model must be updated offline using newly acquired data for training, in order to adapt to the changed dynamics. This is a trade-off that is often present in real-time monitoring systems, where the need for accurate predictions must be balanced with the need for timely updates.

We first demonstrate the performance of the Neural EKF framework for predicting the dynamic response through a numerical example. In this example, we consider a 2 degrees-of-freedom (DOF) nonlinear duffing oscillator subjected to random excitations. The data used for this case study are simulated by the following differential equation:

where $\mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}$, the mass matrix $\mathbf{M}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$, the stiffness matrix $\mathbf{K}=\begin{bmatrix}4&-0.5\\ -0.5&4\end{bmatrix}$, the damping matrix $\mathbf{C}=\begin{bmatrix}0.5&0\\ 0&0.5\end{bmatrix}$ and the coefficient of the cubic stiffness term $k_{n}=1$. Here the displacements of both DOFs $x_{1}$ and $x_{2}$ are assumed available as measurements. It is noted that this parameter setting falls into the scenario where the Duffing oscillator presents a single stable equilibrium point at the origin. We consciously make this choice, as we here adopt this as a nonlinear example, in order to demonstrate the capability of the Neural EKF to deal with nonlinearity, but not rare instabilities or chaotic behavior, which would require a more complex treatment.

For this numerical case study, a number of 1000 random trajectories are generated with both transition and observation noise variances equal to 0.001 for training by the simulated 2-DOF nonlinear Duffing system shown in Eq.(28). Another 5 randomly generated trajectories are used as an unknown dataset for testing the prediction capability of the Neural EKF framework. The prediction results for the free vibration case are shown in Figure 4. As indicated by the results, the Neural EKF is capable of producing satisfactory predictions that reasonably match the reference well for this simulated nonlinear system. As comparison, which is already shown in Fig 3, it is observed that the DMM, which is a pure VAE-based method, fails to capture the dynamics and the predictions are much less accurate.

To further demonstrate the robustness of the proposed model, we have generated artificial observations (measured data) with different levels of noise contamination. The data is generated by Eq.(28) using the same initial values across three cases, but with different levels of additive Gaussian observation noise. More specifically, we generate three different simulated cases corresponding to an observation noise variance of 0.001, 0.01 and 0.1, respectively. The “process noise” is set to be zero, when generating artificial observation data, since integration of a larger transition process noise would cause the solution of the ODE to diverge due to nonlinearity of the Duffing system. The root mean squared error between the predicted and true (noise-free) system response is reported in Table 1. Our results indicate that the difference between the predicted and noise-free system response, for training and prediction under assumed contaminated measurements, are similar across different noise covariance levels. The proposed method thus proves particularly robust to noise contamination, as revealed by the results shown in Figure 5 for the case with different observation noise variances. As comparison, the DMM-predicted results are plotted in dark green, which deviate significantly from the ground-truth and fail to provide accurate and reliable predictions.

The framework is further investigated using real-world seismic data obtained from the Center for Engineering Strong Motion Data (CESMD) haddadi2008center for a 6-story hotel building in San Bernardino, California. It is a mid-rise concrete building installed with a total of nine accelerometers on the first, third and roof floors in different directions. The sensor placement is shown in Figure 6, where three accelerometers are mounted on each of the ground floor, third floor, and roof. The sensors have recorded multiple seismic events from 1987 to 2021. 17 corresponding structural response and ground motion datasets are used to train the Neural EKF with one dataset reserved for evaluating prediction performance, which corresponds to the San Bernardino earthquake in 2009. The data we used are extracted from sensors 1, 4 and 7, which lie along the same direction. The data from sensor 1 are used as the input ground motion, while the data from sensors 4 and 7 are used as output responses. Due to the inconsistency of the sample frequencies, we first pre-process the data using resampling techniques. The signals are uniformly resampled at 50 Hz (the downloaded data was already pre-processed by the website with noise filtering, baseline and sensor offset corrections).

After training the Neural EKF model with a pre-processed training dataset, the predictions on another independent test dataset are obtained by simply feeding the new seismic inputs into the trained model. The predictions and ground truth values are compared in Figure 7. A strong agreement between prediction responses and ground truth responses can be observed. While the data are sampled at different frequencies and earthquakes vary widely in magnitudes, directions, and frequencies, the Neural EKF essentially learns dynamics of the building, thus generating highly consistent predictions.

Among various application fields of Structural Health Monitoring (SHM), wind turbines are gaining increased attention due to their critical significance and competitiveness as a major source of renewable energy. To further demonstrate the value of the framework in SHM for assessing performance and assisting decision making, we validate use of the Neural EKF for vibration monitoring of operational wind turbines. The data used in this paper were obtained and illustrated in ou2021vibration by experimentally testing a small-scale wind turbine blade, as shown in Figure 8, in both healthy and damaged states under varying environmental temperature conditions. All the testing cases are listed in Table 13. Multiple accelerometers and strain gauges are implemented on different positions of the wind turbine. We only make use of accelerometer measurements as the system outputs and the external forces measured at the force transducer $f_{1}$ as the system input to conduct a vibration-based assessment in this case study. The sensor configuration is shown in Figure 9, where the red dots indicate the positions of accelerometers $a_{i}\quad(i=1,2,...,8)$, with $i$ denoting the label of each sensor. The measured signals are low-pass filtered with a cut-off frequency of 380 Hz.

Given that both the transition and observation models are constructed using neural networks, there exist non-unique combinations of these two models that can comparably produce reasonable system responses. It is possible to achieve a good output response with incorrect dynamics and an appropriate selection of parameters $\mathbf{Q}$ and $\mathbf{R}$, which is in fact the philosophy of Kalman filters. $\mathbf{Q}$ and $\mathbf{R}$ represent the process and measurement noise covariance matrices, which are meant to account for modeling and measurement noise, but which nonetheless necessitate a fairly competent process and measurement model for successful operation. The difference in a Deep version of such a Bayesian filter (as the Neural EKF) is that the state-space equations (process and measurement equations), as well as the noise sources, are not prescribed, but learned. This is why the dynamics, as stated in the title, is actually learnable. This is a main difference with respect to standard Bayesian filters, where these parameters, including the noise covariances, need to be preset. As in conventional filtering, here as well, different combinations of transition and observation models with different $\mathbf{Q}$ and $\mathbf{R}$ matrices exist that can accurately reconstruct the response; the algorithm is searching for a local optimum that represents one of these solutions. This implies that the transition model may not be aligned with the actual dynamics of the system, but rather a nonlinear transformation of it, as the observation model is not explicitly defined.

Figure 15 illustrates the rotated learned latent states by identifying an optimal linear relationship with the true system states. The rotated latent states exhibit a similar pattern to the true system states, albeit with presence of some stretching. This is because the relationship between learned and true states is not a simple linear relationship, but rather a nonlinear transformation. In contrast, the DMM-predicted latent states are plotted on the bottom and exhibit distinctly different patterns to the true system states, even after optimal rotation.

When inference of such true latent states is of interest, we suggest the use of a continuous model, such as Neural Ordinary Differential Equations (Neural ODEs¹⁸), which allows to directly obtain the derivative of the latent state variables. Neural ODEs parameterize the derivative of the latent state (vector field) using a neural network. The output of the network is computed using a black-box differential equation solver. Neural ODEs have proven to be particularly useful for continuous-time latent variable models and can be used to model continuous dynamics, thereby enabling the introduction of derivative relations in the latent space and structure our observation model. We do not illustrate more on this approach here, since this comprises an extension which is beyond the objective of this work, which aimed to illustrated the benefits of the EKF structure in learning dynamical systems. We note $\mathbf{z}=[z_{1},z_{2}]$, where $z_{1}$ and $z_{2}$ represent the first and second half elements of the latent state, respectively. Similar to the formulations in Eq.(2) and Eq.(3), but now set in a continuous form with the Neural ODE modeling the transition function, the dynamics can be modeled as:

The first set of equations serves for enforcing the second half of the latent state to assume the form of the derivatives of the first half, while the second equation models the dynamics. Again, this is only when it is of interest to specifically infer a physical latent space. Since the second set of equations is usually chosen to generate a system’s accelerations (as derived from a physics-based equation of motion), it is possible in this case to eliminate the requirement for an additional neural network, for example $g$ in Eq.(3), for modeling the observation process, and the observation function is merely$f$ in this case, as indicated in Eq.(31). However, in this case the purpose of the network is modified, since the idea here is to impose some sort of physics-based structure, as opposed to more freely learn the unknown underlying dynamics.

As described in the theoretical section, the covariance matrices $\mathbf{Q}$ and $\mathbf{R}$ are assumed diagonal, thus for the duffing oscillator example, these are represented by $\mathbf{Q}=\text{diag}(q_{1},q_{2},q_{3},q_{4})$ and $\mathbf{R}=\text{diag}(r_{1},r_{2})$, respectively. The estimated diagonal elements are listed in Table 2.

It is observed that 1) the estimated values of $\mathbf{Q}$ are close to zero and much smaller than the defined values of $\mathbf{R}$, as no process noise was in this example added to the simulated data. 2) Without structuring the observation model, both transition and observation models are parameterized by neural networks, jointly optimized to reconstruct system response. Thus, it is not surprising that the estimated $\mathbf{Q}$ and $\mathbf{R}$ are not exactly matching the true value, but one can still observe that the estimated values of $\mathbf{R}$ increase when larger observation variances are added to the simulated data, thereby accounting for increasing uncertainty in the measured data. It is important to underline that, unlike the conventional EKF, the proposed Neural EKF is a deep learning-based method, meant to learn (infer) the transition and emission models from data. The results in Figure 5 show that even with highly noisy data, Neural EKF is able to recover the true response with high confidence, thus the estimated $\mathbf{R}$ is observed to be smaller. Using the estimated standard deviations, which are the square roots of the estimated variances, 99.7% confidence intervals for predicted responses can be calculated. These intervals should be highly precise around mean values, considering the extremely low values of estimated variances.