# Bayesian physics-informed neural network for parameter estimation of mooring line

Nikhil Mahar, Subhamoy Sen, Laurent Mevel

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Nikhil Mahar, Subhamoy Sen, Laurent Mevel. Bayesian physics-informed neural network for parameter estimation of mooring line. COMSYS-2024 - 5th International Conference on Frontiers in Computing and Systems, Dec 2024, Goa, India. hal-05484453

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# Bayesian physics-informed neural network for parameter estimation of mooring line

Nikhil Mahar<fnref n="1" />, Subhamoy Sen<fnref n="1" />, and Laurent Mevel<fnref n="2" />

**Abstract.** Physics-informed Neural Networks (PINN), embedding partial physics through minimizing residuals of ordinary/partial differential equations (ODEs/PDEs), show promise in reducing the reliance on extensive datasets for training by incorporating physical principles. However, their limitation in providing only point estimates hinders real-life applications where uncertainties are unavoidable. Addressing this, Bayesian Physics Informed Neural Networks (BPINN) with Hamiltonian Monte Carlo (HMC) optimization are employed here for uncertainty quantification in parameter and state estimation. BPINN, known for its versatility, provides interpretable solutions and robust parameter identification, making it particularly valuable in ocean engineering, where addressing significant uncertainties in system estimation poses a major challenge. This study applies BPINN to predict motion and identify system parameters in offshore structures, focusing on mooring lines. Using a simplified dynamics model, the research aims to predict the displacement of the platform surge and reveal the parameters of the system related to the degradation of the mooring line. Incorporating uncertainty quantification enhances the prediction reliability for real-world applications prone to environmental noise and uncertainties.

**Keywords:** Mooring line · PINN · BPINN · System Identification · HMC

## 1 Introduction

Buoys are pivotal in maritime settings, serving not only as navigational aids, but also as sophisticated platforms for real-time observation of oceanic and meteorological data. These observations are instrumental in weather prediction, climate monitoring, and understanding various marine processes. In tandem with the growing prominence of renewable energy, buoys have evolved to harness wave energy, becoming intelligent autonomous platforms that integrate energy extraction, environmental monitoring, and communication systems.

Ensuring the reliability and operational efficiency of buoys in underwater environments is paramount due to their extended operational lifespans. Accurate motion estimation is of significant importance for predicting energy conversion,

<footnote n="1">¹ Indian Institute of Technology Mandi, India</footnote>

<footnote n="2">² Univ. Gustave Eiffel, Inria, Cosys-SII, I4S, Campus de Beaulieu, Rennes, France
subhamoy@iitmandi.ac.in</footnote>
performing failure analysis, and facilitating effective motion planning and con-
trol [4]. However, accurately predicting buoy motion is challenging due to the
intricate interplay between multiple factors such as waves, currents, and winds.
Existing model-based approaches, which incorporate hydrodynamics [9,10] and
environmental effects, have been constrained by the inherent complexity and
insufficient experimental data.

Machine learning (ML) has shown promise in empirically estimating buoy motion [6]. These methods establish complex relationships between environmental factors and motion patterns or make temporal predictions based on motion data. Despite their effectiveness, these data-driven approaches lack interpretability and rely heavily on limited field data.

As buoys are designed for extended operational periods without frequent human intervention, ensuring structural health monitoring (SHM) becomes imperative [3]. Traditional model-based SHM encounters challenges in addressing the intricacies of sea states. Conversely, data-driven ML models offer advantages in discerning changes in the buoy's system through observed data [14]. An optimal approach involves synergizing the benefits of data-driven modeling, particularly harnessing the capability of neural networks to capture nonlinear phenomena, with a comprehensive grasp of the underlying physical principles. The integration of motion estimation into SHM methodologies holds significant promise for substantially augmenting the stability and operational durability of buoys.

Physics-informed neural networks (PINN) have emerged as a prominent data-
driven framework, particularly effective in solving PDEs [8]. Unlike conventional
data-driven methods, PINN embeds the fundamental physics into a residual
neural network, going beyond a mere mapping of labeled data. Its distinctive
approach involves minimizing a weighted sum of PDE residuals, encompassing
boundary and initial conditions. Remarkably, PINN requires minimal adjust-
ment to its loss function, making it versatile for motion estimation and system
parameter identification.

PINNs have demonstrated their versatility, finding applications across vari-
ous engineering domains, spanning power systems, damage mechanics, medical
science, and ocean engineering. In the field of oceanography, PINNs have made
significant contributions to fluid mechanics [2], wave modeling [12], and param-
eter identification [5]. Notable for providing meaningful and interpretable solu-
tions, PINNs excel in efficiently identifying system parameters, even with limited
observational data. This unified framework adeptly addresses both forward and
inverse problems, tackling equation responses and parameter identification with
minimal adjustments in the loss function, showcasing adaptability across diverse
engineering domains. However, the effectiveness of PINN is dependent on the
quality of the available data and the understanding of underlying physics. Real-
world data inherently carry uncertainties and noise, necessitating thoughtful
interpretation within a modeling framework designed to quantify uncertainties.

A promising recent strategy involves integrating PINN into Bayesian statis-
tics. This novel approach merges Bayesian neural networks with PINN to es-
tablish prior distributions and employs Hamilton Monte Carlo methods to esti-
mate posterior distributions [13]. This combination produces credible intervals,
providing essential insights into quantifying uncertainty—an integral aspect of
supporting confidence in model predictions.

Bayesian statistics have gained considerable traction over the years, encour-
aging the development of probabilistic programming and opening new domains
to address a series of problems [1]. The fusion of Bayesian statistics with PINN
towards developing a Bayesian-PINN or BPINN has exhibited remarkable per-
formance across diverse engineering phenomenon.

Inspired by this progress, our study extends the application of BPINN to the
field of ocean engineering. Mooring lines, critical components of offshore struc-
tures subjected to varied sea states, require vigilant health monitoring. In the
current investigation, mooring lines are modeled as nonlinear dynamical sys-
tems, employing the BPINN framework for simultaneous motion and parameter
estimation. This innovative approach offers a comprehensive strategy for moni-
toring and comprehending the dynamics of mooring lines within complex marine
environments.

## 2 Motivation and Contribution

The study aims to improve the reliability of mooring line parameter estima-
tion by integrating Bayesian Physics-Informed Neural Networks (BPINN) with
Hamiltonian Monte Carlo (HMC) optimization for uncertainty quantification.
This approach addresses the limitations of traditional Physics-Informed Neural
Networks (PINN) in handling uncertainties, providing interpretable solutions
and robust parameter identification. The research demonstrates the effective-
ness of BPINN in predicting motion and identifying system parameters in off-
shore structures, specifically focusing on mooring lines, which is crucial for ocean
engineering and other fields where system estimation under uncertainty is chal-
lenging.

3 Problem Statement

The primary focus of this study is the horizontal surge motion of a catenary-
anchored buoy, a subject of considerable interest in various related investigations
[7,10,11]. Figure 1(b) depicts the arrangement of 6 mooring lines anchored to the
cylindrical hollow buoy symmetrically. The system parameters used in this study
are derived from the research presented in [11], as detailed in Table 1.

For simplicity, an idealized nonlinear single degree of freedom system is em-
ployed, as illustrated in Figure 1(c). This model is built on three key assumptions:
(1) the mooring system is represented as a nonlinear SDOF; (2) sea waves are
modeled as a superposition of regular sine waves; and (3) the analysis focuses
solely on surge motion. Specifically, the restoring force $f_{moor}(t)$ is characterized
by a polynomial of 5$^{th}$ order, to capture the non-linearity of the system [7,10].
$$f_{moor}(t) = -(k_1 y + k_2 y^3 + k_3 y^5) \qquad (1)$$

where, $k_1$, $k_2$ and $k_3$ are assumed as $k_1 = 0.03100m$, $k_2 = -0.007864m$, and $k_3 = 0.004607m$ representing the linear and non-linear coefficients of assumed moored system. The equation of motion of non-linear SDOF system representing moored system can be expressed as follows:

$$m\ddot{y} + c\dot{y} + k_1 y + k_2 y^3 + k_3 y^5 = f(t) \qquad (2)$$

where $y$, $\dot{y}$ and $\ddot{y}$ represent displacement, velocity, and acceleration, respectively; the variable $c$ embodies damping coefficients that encompass both structural and hydrodynamic damping, significantly impacting the system's resistance to motion; and the term $f(t)$ represents the wave force acting on the system including both first- and second-order wave forces. Details of the parameters characterizing hydrodynamic force under 5m/5s sea state are shown in Table 2.

<table><caption>**Table 1.** Key parameters of moored system for numerical analysis [11]</caption><thead><tr><th>S.No.</th><th>Moored system parameters</th><th>Value</th></tr></thead><tbody><tr><td>1</td><td>Buoy dimensions (diameter × height × thickness)</td><td>12.5 m × 10 m × 3 m</td></tr><tr><td>2</td><td>Buoy's submerged depth</td><td>4.5 m</td></tr><tr><td>3</td><td>Buoy's mass</td><td>275957.5 kg</td></tr><tr><td>4</td><td>Mooring line mass</td><td>21 kg/m</td></tr><tr><td>5</td><td>Mooring lines</td><td>6</td></tr><tr><td>6</td><td>Sea depth</td><td>125 m</td></tr></tbody></table>

<table><caption>**Table 2.** Details of mass and damping coefficients [11].</caption><thead><tr><th>Sea state</th><th>Added mass (kg)</th><th>Total mass (kg)</th><th>Hydrodynamic damping (Nm/s)</th><th>Total damping (Nm/s)</th></tr></thead><tbody><tr><td>5m/5s</td><td>151163.9</td><td>427121.44</td><td>97604.09</td><td>100762.3</td></tr></tbody></table>

The buoy experiences wave-induced forces calculated using [9]'s methodology, providing a comprehensive solution for both first- and second-order wave diffraction forces acting on a vertical circular cylinder in regular head waves. These forces are expressed as:

$$f_1(t) = A_1 \sin(\omega_f t) \qquad (3)$$

and,

$$f_2(t) = A_2 \sin(\omega_f t) + f_m \qquad (4)$$
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic Representation of the Moored System: (a) Elevation view illustrates the vertical arrangement of the buoy and mooring lines. The buoy, with dimensions of 12.5 m in diameter, 10 m in height, and 3 m in thickness, is submerged to a depth of 4.5 m. The mooring lines extend from the buoy to the seabed, which is 125 m deep; (b) Plan view shows the top-down view of the buoy system, highlighting the symmetrical distribution of six slack mooring lines around the cylindrical buoy; (c) Idealized system depicts the simplified nonlinear oscillator model used for analysis. The model assumes a single-degree-of-freedom system with the mooring lines exerting a restoring force modeled by a fifth-order antisymmetric polynomial function. The equation of motion includes terms for displacement, velocity, acceleration, damping, and wave forces acting on the system</figcaption></figure>

<table><caption>**Table 3.** Wave force characteristics for numerical analysis[11].</caption><thead><tr><th>Sea state</th><th>Frequency of wave<br>ω<sub>f</sub> (rad/s)</th><th>1<sup>st</sup> order force<br>amplitude</th><th>2<sup>nd</sup> order force<br>amplitude</th><th>Steady 2<sup>nd</sup><br>order force</th></tr></thead><tbody><tr><td>5m/5s</td><td>1.2566</td><td>333581.8 N</td><td>16833.41 N</td><td>20834.79 N</td></tr></tbody></table>

Here, $\omega_f$ signifies the frequency of excitation. The 2$^{nd}$ order wave force consists of a constant $f_m$ alongside an oscillating 2$^{nd}$ order force as shown in Table 3

In this study, numerical simulation of mass normalized moored system is done in state space setting for 20 sec with 50Hz sampling frequency. The governing equation of mass normalized system can be expressed as:

$$ \ddot{y} + c_n \dot{y} + k_{1n} y + k_{2n} y^3 + k_{3n} y^5 = f_n(t) \quad (5) $$

where, $c_n = c/m$, $k_{1n} = k_1/m$, $k_{2n} = k_2/m$, $k_{3n} = k_3/m$ with m being the scalar mass value for the SDOF system. Consequently, the wave force after normalization is rendered as $f_n(t) = A_{1n}sin(\omega_f t) + A_{2n}sin(2\omega_f t)$, with normalized parameters $A_{1n} = A_1/m$, $A_{2n} = A_2/m$, and $f_{mn} = f_m/m$.
## 3.1 Bayesian neural network (BNN)

The proposed network connecting the parameters with their respective responses is defined with L layers, with the corresponding weights, biases, and activation functions being $W_l$, $b_l$, and $\sigma_l$ for all $l = 1, \dots, L$. The functional representation can be represented as:

$$ \tilde{y} = \frac{1}{n} \sigma_l (W_l \cdot \sigma_{l-1} (W_{l-1} \cdot \dots \sigma_2 (W_2 \cdot \sigma_1 (W_1 \cdot t + b_1) + b_2) \dots) + b_l) \quad (6) $$

The goal of a typical Bayesian Neural Network (BNN) is the posterior distribution $P(\theta|y)$ estimation of the network parameters $\theta = \{W_l, b_l\}$, given the input $t$, without relying on prior knowledge of the physical system. This ensures that the output of the neural network $\tilde{y}$ aligns with the observed data $y$. Unlike conventional neural networks, in a BNN, the parameters $W_l$ and $b_l$ are not initialized with random point values; instead, they are assigned prior probability distributions, denoted as $P(\theta)$. This initial estimate is then refined by updating the prior using the data's likelihood, $P(y|\theta)$.

$$ P(\theta | y) = \frac{P(y | \theta) P(\theta)}{P(y)} \quad (7) $$

$P(y|\theta)$, the likelihood, is the conditional probability of output data $y$ concerning the network parameters $\theta$. $P(y)$ is the marginal likelihood or evidence. The optimization of the network parameter, $\theta$, is done using HMC sampler.

## 3.2 BPINN

Although a BNN can offer credible intervals in approximating solutions, it demands a comparatively higher amount of data for training. Hence, in this study, BPINN is employed for dual state-parameter estimation instead of a BNN. The need for parameter estimation leads to an inverse estimation framework, where system parameters, such as stiffness and damping, are required in addition to network parameters $\theta$. In this context, a parameter vector $\mu$ (i.e., $\mu = \{k_{1n}, k_{2n}, k_{3n}\}$) is introduced and augmented in the estimable parameter vector $\Phi = \{\theta; \mu\}$, subsequently estimated using the proposed BPINN network. The objective is to ensure that the statistics of the network's predicted output $\tilde{y}$ align with both the statistics of the actual displacement data $y$ and the governing physics represented by $z$ (cf. equation 5). BPINN serves the purpose of estimating the posterior distribution, $P(\Phi|y, z)$.

Mathematically, the formulation is represented as:

$$ P(\Phi | y, z) = \frac{P(y, z | \Phi) P(\Phi)}{P(y) P(z)} = \frac{P(y | \Phi) P(z | \Phi) P(\theta) P(\mu)}{P(y) P(z)} \quad (8) $$

The expression $P(y, z | \Phi) = P(y | \Phi)P(z | \Phi)$ represents the joint likelihood, where $P(y | \Phi)$ is the likelihood of the observed data $y$ given the parameters $\Phi$, and $P(z | \Phi)$ is the likelihood of the physical constraints $z$ given
<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic of the proposed BPINN framework. This diagram illustrates the Bayesian Physics-Informed Neural Network (BPINN) approach for parameter identification in mooring systems. The process involves updating weights and parameters to maximize the joint likelihood of observed data and governing physics, ensuring accurate and reliable system parameter estimation.</figcaption></figure>

the same parameters. The prior probability distribution of $\theta$ and $\mu$ is represented by the term $P(\Phi) = P(\theta)P(\mu)$, whereas evidence, $P(y)P(z)$, denotes the overall probability of data $y$ and the physics $z$. Finally, $P(\Phi | y, z)$ is the posterior distribution, reflecting the updated belief about the parameters $\Phi$ after incorporating both the observed data and the physical constraints.

The likelihood function $P(y | \Phi)$ assesses the alignment between the actual measurement $y$ and the network prediction $\tilde{y}$ at each time step $i$ ($y_i = y(t_i)$). Each individual likelihood function $p_i(y_i | \theta)$ can be chosen as a normal distribution to quantify this alignment. This choice often works well in practice and allows for tractable computations in many scenarios. The overall likelihood can further be estimated as,

$$P(y | \Phi) = \prod_{i=0}^{n} p_i(y_i | \Phi) \quad \text{with} \quad p_i(y_i | \Phi) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left( - \frac{\|\tilde{y}_i - y_i\|^2}{2\sigma^2} \right) \quad (9)$$

$\tilde{y}_i$ represents the network prediction for the variable $y$ at time instant $i$, while $\sigma$ serves as the tuning parameter that allows control over the pragmatic balance between accuracy and the speed of convergence. The second likelihood function $P(z | \Phi)$ assesses the fidelity of the model output $\tilde{y}$ with respect to the governing physics $z$ at each observation time step $i$. In this context, $z_i$ reflects the residual denoting the deviation in the dynamic equation from adherence to physics and is
contingent on the prediction $\tilde{y}_i$. Once again, the likelihood function is expressed as the product of individual likelihood functions $p_i(z_i | \Phi)$. Assuming a normal distribution for each of these likelihood functions, it can be formulated as:

$$P(\mathbf{z} | \mathbf{\Phi}) = \prod_{i=0}^{n} p_i(z | \mathbf{\Phi}) \quad \text{with} \quad p_i(z | \mathbf{\Phi}) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left( -\frac{\|z(t_i)\|^2}{2\sigma^2} \right) \quad (10)$$

The priors for the physics parameters are defined in Equation 11, where the logarithms of $k_{1n}$, $k_{2n}$, and $k_{3n}$ are independently sampled from log-normal distributions. This choice reflects a weakly informative prior, centered around a logarithmic value of 0.05 with moderate variability. For the network architecture, it consists of two hidden layers ($L = 3$ in total) with 20 neurons each, as well as an input layer and a single-neuron output layer. The network parameters $\mathbf{W}_l$ and $\mathbf{b}_l$ are drawn from a normal distribution, as specified in Equation 12.

$$\begin{align*}
\log(k_{1n}) &\sim \mathcal{N}(\mu = \log(0.035), \sigma = 0.05) \\
\log(k_{2n}) &\sim \mathcal{N}(\mu = \log(0.007), \sigma = 0.05) \\
\log(k_{3n}) &\sim \mathcal{N}(\mu = \log(0.0055), \sigma = 0.05)
\end{align*}
\qquad (11)$$

$$\mathbf{W}_l, \mathbf{b}_l \sim \mathcal{N}(\mu = 0.0, \sigma = 2.0) \qquad (12)$$

These prior distributions encapsulate initial beliefs about the range and variability of physics and network parameters before observing any data. The strongly informed priors provide flexibility in model fitting while incorporating preliminary expectations about parameter values. Posterior distributions are approximately inferred through Bayes' theorem in python Tensorflow-Probability using HMC sampler which involves 50 leapfrog steps, and an initial time step of $dt = 5e-4$. The initial 5000 samples are utilized for sampler tuning, while the subsequent 3000 samples are employed for estimating parameters. BPINN-based training has been performed in the Python environment using Tensorflow on a computer with 12 Intel Core i7-12700K CPUs. All the codes will be available in Github repository after publication. The overall methodology adopted for parameter estimation and response reconstruction is illustrated in Figure 3.

# 4 Result and Discussion

The proposed BPINN framework is applied for simultaneous estimation of both network and system parameters in a non-linear SDOF system. The SDOF system is simulated, and sparsely sampled data is utilized for parameter estimation. Remarkably, despite using a sparse dataset of only 100 data points, BPINN successfully reconstructs and extrapolates the system's response, as demonstrated in Figure 4. The reconstructed displacement profile shows the model's ability to capture the system's dynamics with high accuracy.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Methodology for parameter estimation and response reconstruction. This schematic illustrates the process of parameter estimation and response reconstruction using the BPINN framework. The mooring system simulation generates sparse data, which is then processed by the BPINN to probabilistically estimate parameters and reconstruct the system's response.</figcaption></figure>

Figure 5 presents the estimated mean values of the system parameters, which are 0.033, 0.0077, 0.0047, compared to the true values 0.031, 0.0079, 0.0046. The associated standard deviations for the parameters are 0.0015, 0.0002, 0.0002, illustrating the model's precision. These results indicate that BPINN not only captures the underlying system parameters with high accuracy but also provides valuable uncertainty quantification through the standard deviations of the estimates.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Displacement prediction of the mooring system. The model accurately reconstructs and extrapolates the system's motion even with sparse data, demonstrating the robustness and reliability of the BPINN approach in capturing the dynamics of the mooring line.</figcaption></figure>

Table 4 presents a comparative analysis between the proposed Bayesian Physics-Informed Neural Network (BPINN) and the conventional Physics-Informed Neural Network (PINN) approach. The results clearly demonstrate that BPINN significantly outperforms the conventional PINN in terms of parameter estimation accuracy. For instance, BPINN yields a 6.45% error in estimating the parameter $k_{1n}$, whereas PINN incurs a much higher error of 38.7%, showcasing the enhanced precision of BPINN.

To further evaluate the robustness of BPINN, non-informative physical priors are employed as described in Equation 13. The mean estimated values for the system parameters {0.026, 0.002, 0.0013} are compared to the true values
<figure><img src="image_6.png" /><figcaption>Fig. 5. Estimated parameters of mooring system. The model estimated posterior distribution of unknown parameters $k_{1n}$, $k_{2n}$, and $k_{3n}$ even with sparse data, demonstrating the robustness and reliability of the BPINN approach.</figcaption></figure>

$\{0.031, 0.0079, 0.0046\}$. This comparison reveals the limitations of BPINN in accurately capturing the true dynamics of the system when using non-informative priors. Consequently, future research may focus on employing non-dimensionalized systems, which could potentially lead to improved parameter estimation accuracy.

$$ 
\begin{aligned}
\log(k_{1n}) &\sim \mathcal{N}(\mu = \log(1), \sigma = 0.05) \\
\log(k_{2n}) &\sim \mathcal{N}(\mu = \log(1), \sigma = 0.05) \\
\log(k_{3n}) &\sim \mathcal{N}(\mu = \log(1), \sigma = 0.05)
\end{aligned} 
\qquad (13) $$

This study showcases the robustness of the BPINN framework, which effectively integrates physical laws and Bayesian inference to estimate system parameters with limited training data. The inclusion of Bayesian techniques not only improves the accuracy of the parameter estimation but also provides a natural framework for uncertainty quantification, as demonstrated by the posterior distributions of the parameters. This ability to quantify uncertainty is crucial for improving the reliability and interpretability of the model, especially in scenarios with sparse data.

<table><caption>Table 4. Comparison of estimated morning line parameters between proposed BPINN and existing PINN [5]</caption><thead><tr><th rowspan="2">Parameter<br>k<sub>in</sub></th><th rowspan="2">Truth</th><th colspan="3">BPINN</th><th colspan="3">PINN [5]</th></tr><tr><th>E[k<sub>in</sub>]</th><th>std[k<sub>in</sub>]</th><th>Error (%)</th><th>E[k<sub>in</sub>]</th><th>std[k<sub>in</sub>]</th><th>Error (%)</th></tr></thead><tbody><tr><td>i=1</td><td>0.0310</td><td>0.0330</td><td>0.0015</td><td>6.45</td><td>0.0430</td><td>0.0660</td><td>38.70</td></tr><tr><td>i=2</td><td>0.0079</td><td>0.0077</td><td>0.0002</td><td>2.53</td><td>0.0078</td><td>0.0833</td><td>1.26</td></tr><tr><td>i=3</td><td>0.0046</td><td>0.0047</td><td>0.0002</td><td>2.17</td><td>0.0046</td><td>0.0470</td><td>0.0</td></tr></tbody></table>
# 5 Conclusion

The study implemented a Bayesian extension of the state-of-the-art Physics Informed Neural Network (PINN) architecture. The objective was to tackle real-life uncertainties using data-driven methods, which traditionally provide only point estimates. The incorporation of an inverse framework within the PINN architecture allows for the integration of system physics into the data-driven estimation approach, while the Bayesian extension introduces a more practical treatment of real-life system parameters as probabilistic random variables. The primary investigation focused on a simple SDOF system to assess the relative potential of the proposed approach. Future work will involve extending the methodology to handle more complex real-life MDOF systems.

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