A Sobolev-Information Perspective on Derivative-Observation-Augmented PINNs for Parameter Identification of Second-Order Dynamical Systems

1. Introduction

Second-order dynamical systems governed by ordinary differential equations of the form $m\ddot{u}+\mu \dot{u}+ku=f\left(t\right)$ are ubiquitous in structural dynamics, mechanical engineering, and vibration analysis [1,21]. Accurate identification of system parameters, particularly the damping coefficient $\mu$, from limited measurement data is essential for structural health monitoring, fault diagnosis, and model predictive control [24,25].

Traditional parameter estimation methods such as nonlinear least squares (NLS) and Kalman filtering can achieve high accuracy when measurement data are abundant and the system model is exactly known [22]. However, these methods rely on repeated numerical integration of the governing equations, which becomes computationally expensive for nonlinear systems and cannot easily incorporate additional physical constraints beyond the forward model.

Physics-informed neural networks (PINNs), introduced by Raissi et al. [1], offer an alternative paradigm by embedding differential equation residuals as soft constraints in neural network training. For inverse problems, PINNs can simultaneously approximate the solution function and estimate unknown parameters from sparse observations [4,7]. This is particularly attractive for engineering applications where sensor deployment is costly and data are scarce [8].

However, PINN-based parameter estimation for second-order ODEs faces a fundamental difficulty that has received limited theoretical attention: the degeneracy of the inverse problem under displacement-only observations. Specifically, for a homogeneous underdamped oscillator $m\ddot{u}+\mu \dot{u}+ku=0$, different values of $\mu$ can produce displacement responses $u\left(t\right)$ that differ only subtly in their decay envelope, and are effectively indistinguishable when sampled at a sparse set of time points [23]. The PINN loss landscape in such cases admits a manifold of observationally near-equivalent parameter values, making the optimization ill-conditioned regardless of network capacity or training algorithm [5].

This degeneracy is not merely an optimization difficulty but reflects a genuine lack of information in the displacement observations alone. To resolve it, one must supplement the observational data with additional measurements that are sensitive to the parameter of interest. In second-order systems, the velocity $u^{\prime }\left(t\right)$ is directly sensitive to the damping coefficient $\mu$ through the term $\mu u^{\prime }\left(t\right)$, making velocity measurements an information-theoretically natural complement to displacement observations.

Motivated by this observation, we propose derivative-observation-augmented PINNs (D-PINN), which explicitly incorporate first-order derivative (velocity) observations $\left\{u^{\prime }\left(t_j\right)\right\}$ into the PINN loss function. The key contributions of this work are:

• Theoretical analysis of the information gain from derivative observations. We prove three results: (i) a Sobolev-type inequality showing that derivative error controls displacement error when initial conditions are enforced (Theorem 1); (ii) a Fisher information decomposition proving that derivative observations add nonnegative information and strictly improve local parameter identifiability when velocity sensitivity is nonzero (Theorem 2); and (iii) a residual-based parameter stability estimate bounding the parameter error by the physical residual and its derivatives (Theorem 3).
• Comprehensive numerical validation with reproducible experimental design. We evaluate D-PINN against eight baselines (nonlinear least squares, standard PINN, Fourier-feature PINN, gradient-enhanced PINN, Sobolev-PINN, and D-PINN variants with and without Fourier features and parameter priors) on three dynamical systems: linear underdamped oscillator, forced vibration near resonance, and nonlinear Duffing oscillator. All experiments use 10 random seeds and report mean ± standard deviation.
• Characterization of applicability and limitations. We clarify that the primary benefit of D-PINN lies in parameter estimation rather than displacement fitting, as standard PINNs already achieve low displacement mean squared error (MSE) in many settings. We further identify the conditions under which derivative observations improve parameter estimation and examine how measurement quality, including exact derivatives, noisy derivatives, finite-difference approximations, and smoothed finite-difference approximations, affects performance.

The remainder of this paper is organized as follows. Section 2 reviews related work. Section 3 formulates the inverse problem and identifies the parameter degeneracy. Section 4 presents the D-PINN method. Section 5 provides the theoretical analysis. Section 6 reports numerical experiments. Section 7 discusses limitations and Section 8 concludes.

2. Related Work

2.1. Physics-Informed Neural Networks for Inverse Problems

Since their introduction by Raissi et al. [1], PINNs have been applied to inverse problems across fluid mechanics [2], nano-optics [3], heat conduction [20], and structural dynamics [16,17]. In the inverse setting, unknown parameters are treated as trainable variables alongside the network weights, and the physics residual provides the supervisory signal for parameter learning [7]. However, the parameter estimation accuracy of standard PINNs is often unsatisfactory when observation data are sparse, because the physics residual alone provides a weak constraint that is insufficient to uniquely determine the parameters [4,8].

2.2. Sobolev Training and Derivative Supervision

The idea of training neural networks with derivative information originates from Sobolev training [15], where the network is trained to match not only function values but also their derivatives. In the PINN context, this principle has been extended in several directions. Gradient-enhanced PINNs (gPINN) [6] penalize the gradient of the physics residual ${\nabla }_{\mathbf{x}}\mathcal{R}\left(\mathbf{x}\right)$, which encourages spatial smoothness of the residual but does not incorporate observational derivative data. In contrast, D-PINN directly supervises derivative observations, providing information about the local dynamic evolution that the global physics residual cannot supply. The distinction between residual gradient regularization and derivative observation supervision is crucial and is analyzed in detail in Section 5.

2.3. Gradient-Enhanced and Adaptive-Loss PINNs

Several methods have been proposed to improve PINN training dynamics. Adaptive activation functions [9] and learning rate annealing [5] address gradient pathologies in stiff PDEs. Adaptive collocation point sampling [10,11] places more residual points in high-error regions. Loss balancing methods such as ReLoBRaLo [12] and self-adaptive PINNs [13] dynamically adjust loss weights during training. While these methods improve convergence, they do so by using the available information more effectively rather than by introducing additional information into the inverse problem. D-PINN, by contrast, adds genuinely new information via derivative observations.

2.4. PINNs for Vibration and Second-Order Dynamical Systems

Applications of PINNs to vibration problems include beam vibration modeling [16], bridge damping ratio identification [17], and coal stick-slip impact analysis [18]. Wang et al. [8] studied parameter identification of vibration systems using PINNs under sparse data and identified the under-constrained nature of the problem, but did not propose derivative observations as a solution. In the frequency domain, FD-PINN [19] transforms the governing equations to avoid time integration, but this approach is not directly applicable to nonlinear systems.

3. Problem Formulation

3.1. Second-Order Dynamical System

We consider a general second-order ordinary differential equation governing the displacement $u\left(t\right)$ of a single-degree-of-freedom oscillator:

$$m{\displaystyle \frac{d^{2}u}{d{t}^2}}+\mu {\displaystyle \frac{du}{dt}}+ku=f\left(t\right),t\in [0,T],$$

(1)

with initial conditions $u\left(0\right)=u_0$, $u^{\prime }\left(0\right)=v_0$, where $m>0$ is the mass, $\mu >0$ is the damping coefficient, $k>0$ is the stiffness, and $f\left(t\right)$ is a known external forcing function. For the homogeneous case ($f\left(t\right)\equiv 0$) with $u_0=1$, $v_0=0$, and underdamped condition $\delta =\mu /\left(2m\right)<{\omega }_0=\sqrt{k/m}$, the exact solution is

$$u\left(t\right)=e^{-\delta t}\left[cos\left(\omega t\right)+{\displaystyle \frac{\delta }{\omega }}sin\left(\omega t\right)\right],\omega =\sqrt{{\omega }_0^2-{\delta }^2}.$$

(2)

We also consider two extensions: (i) forced vibration where $f\left(t\right)=F_{0}sin\left({\omega }_{f}t\right)$ with near-resonance forcing (${\omega }_f\approx {\omega }_0$); and (ii) the Duffing oscillator where a cubic nonlinearity is added: $f\left(t\right)=-\alpha {\left[u\left(t\right)\right]}^3$.

3.2. Inverse Problem Statement

The inverse problem is: given a finite set of noisy displacement observations ${\mathcal{D}}_u={\left\{(t_i,{\widehat{u}}_i)\right\}}_{i=1}^{N_u}$ where ${\widehat{u}}_i=u\left(t_i\right)+{\epsilon }_i$ with ${\epsilon }_i\sim \mathcal{N}(0,{\sigma }_u^2)$, and optionally a set of derivative observations ${\mathcal{D}}_v={\left\{(s_j,{\widehat{v}}_j)\right\}}_{j=1}^{N_v}$ where ${\widehat{v}}_j=u^{\prime }\left(s_j\right)+{\eta }_j$ with ${\eta }_j\sim \mathcal{N}(0,{\sigma }_v^2)$, estimate the unknown damping coefficient $\mu$ assuming m and k are known.

3.3. Parameter Degeneracy Under Sparse Displacement-Only Observations

A key difficulty of this inverse problem is that the mapping $\mu \mapsto u(·;\mu )$ is not injective on finite sample sets when only displacement is observed. For the homogeneous underdamped oscillator with parameters $m=1$, $k=400$, and ${\mu }^{*}=4.0$, the displacement responses generated by $\mu =3$ and $\mu =4$ differ by a maximum of only $0.04$ over $t\in [0,1]$, with the mean absolute difference being $0.016$. At 20 uniformly spaced observation points, a ${\chi }^2$ test cannot reliably distinguish these two models given typical measurement noise. This creates a manifold of observationally near-equivalent parameter values in the loss landscape, making gradient-based optimization prone to parameter bias.

In contrast, the velocity responses $u^{\prime }(t;\mu )$ vary substantially with $\mu$. The maximum difference between $\mu =3$ and $\mu =4$ is approximately 0.18, nearly five times the corresponding displacement difference. This suggests that velocity observations provide substantially more information about $\mu$ than displacement observations alone.

4. Derivative-Observation-Augmented PINN

4.1. Standard PINN for Second-Order ODE Inverse Problems

A standard PINN for the inverse problem employs a neural network $u_{\theta }\left(t\right)$ parameterized by weights $\theta$ and a trainable damping coefficient ${\mu }_{\theta }=exp(log{\mu }_{\theta })$ (log-parameterized for positivity). The training objective minimizes a composite loss function:

$${\mathcal{L}}_{\mathrm{PINN}}={\lambda }_{\mathrm{phys}}{\mathcal{L}}_{\mathrm{phys}}+{\lambda }_{\mathrm{data}}{\mathcal{L}}_{\mathrm{data}}+{\lambda }_{\mathrm{init}}{\mathcal{L}}_{\mathrm{init}},$$

(3)

where the physics residual loss, data loss, and initial condition loss are respectively

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{phys}}& ={\displaystyle \frac{1}{N_c}}{\displaystyle \sum _{i=1}^{N_c}}{\left[m{u}_{\theta }^{\prime \prime }\left(t_i\right)+{\mu }_{\theta }{u}_{\theta }^{\prime }\left(t_i\right)+k{u}_{\theta }\left(t_i\right)-f\left(t_i\right)\right]}^2,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{data}}& ={\displaystyle \frac{1}{N_u}}{\displaystyle \sum _{j=1}^{N_u}}{\left[u_{\theta }\left(t_j\right)-{\widehat{u}}_j\right]}^2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{init}}& ={\left[u_{\theta }\left(0\right)-u_0\right]}^2+{\left[u_{\theta }^{\prime }\left(0\right)-v_0\right]}^2.\hfill \end{array}$$

(6)

The collocation points ${\left\{t_i\right\}}_{i=1}^{N_c}$ for the physics residual may differ from the observation points and are typically sampled uniformly from the time domain. The derivatives $u_{\theta }^{\prime }$ and $u_{\theta }^{\prime \prime }$ are computed via automatic differentiation.

4.2. Derivative Observation Loss

D-PINN augments the standard PINN loss with a derivative observation term:

$${\mathcal{L}}_{\mathrm{deri}}={\displaystyle \frac{1}{N_v}}{\displaystyle \sum _{j=1}^{N_v}}{\left[u_{\theta }^{\prime }\left(s_j\right)-{\widehat{v}}_j\right]}^2.$$

(7)

The total D-PINN loss is

$${\mathcal{L}}_{\mathrm{D}-\mathrm{PINN}}={\lambda }_{\mathrm{phys}}{\mathcal{L}}_{\mathrm{phys}}+{\lambda }_{\mathrm{data}}{\mathcal{L}}_{\mathrm{data}}+{\lambda }_{\mathrm{deri}}{\mathcal{L}}_{\mathrm{deri}}+{\lambda }_{\mathrm{init}}{\mathcal{L}}_{\mathrm{init}}.$$

(8)

The derivative observation loss directly constrains the local gradient of the learned solution at the observation times. This is fundamentally different from gPINN [6], which penalizes ${\nabla }_t\mathcal{R}\left(t\right)$ (the gradient of the residual) rather than matching observed derivatives. The gPINN residual gradient penalty encourages smoothness of the residual function, whereas D-PINN’s derivative observation loss injects external measurement information.

4.3. Fourier Feature Representation

To improve the representation of high-frequency oscillatory behavior, we apply a deterministic Fourier feature mapping [14] to the input:

$$\mathrm{\Phi }\left(t\right)={\left[t,sin\left(2\pi f_{1}t\right),cos\left(2\pi f_{1}t\right),\dots ,sin\left(2\pi f_{N}t\right),cos\left(2\pi f_{N}t\right)\right]}^{\top },$$

(9)

with frequencies $f_i=2^{i-1}$ for $i=1,\dots ,6$. The network architecture is a 4-layer fully connected network with 128 neurons per layer and tanh activation. The trainable parameter ${\mu }_{\theta }$ is embedded as a log-parameterized variable to ensure positivity.

4.4. Non-Oracle Parameter Regularization

To incorporate coarse prior knowledge without leaking the true parameter value, we optionally add a weak regularization term:

$${\mathcal{L}}_{\mathrm{reg}}={\lambda }_{\mathrm{reg}}{({\mu }_{\theta }-{\mu }_{\mathrm{prior}})}^2,$$

(10)

where ${\mu }_{\mathrm{prior}}$ is a coarse estimate (e.g., from engineering experience or a preliminary identification [22]) and ${\lambda }_{\mathrm{reg}}$ is a small regularization weight. Critically, ${\mu }_{\mathrm{prior}}$ must not equal the true parameter value ${\mu }^{*}$ in the main experimental evaluation; using ${\mu }_{\mathrm{prior}}={\mu }^{*}$ constitutes an oracle prior and is reported only as a diagnostic reference in the Appendix. In the main experiments, we use either ${\lambda }_{\mathrm{reg}}=0$ (no prior) or ${\lambda }_{\mathrm{reg}}=0.01$ with priors ${\mu }_{\mathrm{prior}}\in \{3.2,4.8,2.5,5.5\}$ that deviate from the true value ${\mu }^{*}=4.0$ by $20\%$ to $37.5\%$.

4.5. Training Procedure

The complete D-PINN training objective is

$$\mathcal{L}=5{\mathcal{L}}_{\mathrm{phys}}+10{\mathcal{L}}_{\mathrm{data}}+0.1{\mathcal{L}}_{\mathrm{deri}}+1{\mathcal{L}}_{\mathrm{init}}+{\lambda }_{\mathrm{reg}}{({\mu }_{\theta }-{\mu }_{\mathrm{prior}})}^2.$$

(11)

The loss weights ${\lambda }_{\mathrm{phys}}=5$, ${\lambda }_{\mathrm{data}}=10$, ${\lambda }_{\mathrm{deri}}=0.1$, and ${\lambda }_{\mathrm{init}}=1$ were determined through sensitivity analysis on the derivative loss weight (see Section 6.5). Training uses the Adam optimizer with initial learning rate ${10}^{-3}$ and cosine annealing to ${10}^{-5}$ over 20,000 epochs.

5. Theoretical Analysis

We now provide three theoretical results that explain why derivative observations improve parameter identifiability. All proofs assume the initial condition is exactly enforced, which is a standard capability of PINNs via the ${\mathcal{L}}_{\mathrm{init}}$ term.

5.1. Sobolev-Type Error Control from Derivative Supervision

Theorem 1.

Let $e\left(t\right)=u_{\theta }\left(t\right)-u^{*}\left(t\right)$ be the displacement error, where $u_{\theta }\in C^2\left([0,T]\right)$ and $u^{*}$ is the true solution. Assume $e\left(0\right)=0$ (initial displacement is exactly matched). Then the following Sobolev-type inequalities hold:

$$\begin{array}{cc}\hfill {\parallel e\parallel }_{L^2(0,T)}& \le T\parallel e^{\prime }{\parallel }_{L^2(0,T)},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\parallel e\parallel }_{L^{\infty }(0,T)}& \le \sqrt{T}{\parallel e^{\prime }\parallel }_{L^2(0,T)}.\hfill \end{array}$$

(13)

Proof. 

Since $e\left(0\right)=0$, the fundamental theorem of calculus gives $e\left(t\right)={\int }_0^t{e}^{\prime }\left(s\right)ds$ for any $t\in [0,T]$. By the Cauchy–Schwarz inequality,

$$\left|e\left(t\right)\right|=\left|{\int }_0^t{e}^{\prime }\left(s\right)ds\right|\le \sqrt{t}{\left({\int }_0^t{\left|e^{\prime }\left(s\right)\right|}^{2}ds\right)}^{1/2}\le \sqrt{T}{\parallel e^{\prime }\parallel }_{L^2(0,T)}.$$

Taking the supremum over $t\in [0,T]$ yields (). For the $L^2$ bound, integrate ${\left|e\left(t\right)\right|}^2\le t{\int }_0^t|e^{\prime }{\left(s\right)|}^{2}ds\le t{\parallel e^{\prime }\parallel }_{L^2(0,T)}^2$:

$${\parallel e\parallel }_{L^2(0,T)}^2={\int }_0^T{\left|e\left(t\right)\right|}^{2}dt\le \parallel e^{\prime }{\parallel }_{L^2(0,T)}^2{\int }_0^{T}tdt={\displaystyle \frac{T^2}{2}}{\parallel e^{\prime }\parallel }_{L^2(0,T)}^2,$$

which is a slightly sharper bound than (12); the stated inequality follows from ${\displaystyle \frac{T}{\sqrt{2}}}<T$. □

Interpretation. Theorem 1 establishes that if the derivative error $\parallel e^{\prime }{\parallel }_{L^2}$ is controlled (e.g., through derivative observation supervision), then both the $L^2$ and $L^{\infty }$ displacement errors are automatically bounded. In a standard PINN without derivative observations, $\parallel e^{\prime }{\parallel }_{L^2}$ is only indirectly constrained through the physics residual, which couples e, $e^{\prime }$, and $e^{\prime \prime }$. The D-PINN derivative observation loss provides a direct mechanism to reduce $\parallel e^{\prime }{\parallel }_{L^2}$, thereby tightening the displacement error bound. The constant $T=1$ in our experiments gives ${\parallel e\parallel }_{L^{\infty }}\le {\parallel e^{\prime }\parallel }_{L^2}$, meaning velocity error directly bounds displacement error.

5.2. Fisher Information Gain from Derivative Observations

Theorem 2.

Assume independent Gaussian observation models:

$$\begin{array}{cc}\hfill {\widehat{u}}_i& =u(t_i;{\mu }^{*})+{\epsilon }_i,{\epsilon }_i\sim \mathcal{N}(0,{\sigma }_u^2),i=1,\dots ,N_u,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\widehat{v}}_j& =u^{\prime }(s_j;{\mu }^{*})+{\eta }_j,{\eta }_j\sim \mathcal{N}(0,{\sigma }_v^2),j=1,\dots ,N_v.\hfill \end{array}$$

(15)

The Fisher information for ${\mu }^{*}$ with both displacement and derivative observations is

$${\mathcal{I}}_D\left({\mu }^{*}\right)={\mathcal{I}}_P\left({\mu }^{*}\right)+{\displaystyle \frac{1}{{\sigma }_v^2}}{\displaystyle \sum _{j=1}^{N_v}}{\left({\displaystyle \frac{\partial u^{\prime }(s_j;\mu )}{\partial \mu }}{|}_{\mu ={\mu }^{*}}\right)}^2\ge {\mathcal{I}}_P\left({\mu }^{*}\right),$$

(16)

where ${\mathcal{I}}_P\left({\mu }^{*}\right)={\sigma }_u^{-2}{\sum }_{i=1}^{N_u}{(\partial u(t_i;{\mu }^{*})/\partial \mu )}^2$ is the Fisher information from displacement-only observations.

Proof. 

The log-likelihood for the combined data is

$$\ell \left(\mu \right)=-{\displaystyle \frac{1}{2{\sigma }_u^2}}{\displaystyle \sum _{i=1}^{N_u}}{({\widehat{u}}_i-u(t_i;\mu ))}^2-{\displaystyle \frac{1}{2{\sigma }_v^2}}{\displaystyle \sum _{j=1}^{N_v}}{({\widehat{v}}_j-u^{\prime }(s_j;\mu ))}^2+\mathrm{const}.$$

The Fisher information is $\mathcal{I}\left({\mu }^{*}\right)=-\mathbb{E}[{\partial }^2\ell /\partial {\mu }^2]{|}_{\mu ={\mu }^{*}}$. Computing the second derivative and taking the expectation (the cross-term vanishes because $\mathbb{E}\left[{\epsilon }_i\right]=\mathbb{E}\left[{\eta }_j\right]=0$) yields

$${\mathcal{I}}_D\left({\mu }^{*}\right)={\displaystyle \frac{1}{{\sigma }_u^2}}{\displaystyle \sum _{i=1}^{N_u}}{\left({\displaystyle \frac{\partial u(t_i;{\mu }^{*})}{\partial \mu }}\right)}^2+{\displaystyle \frac{1}{{\sigma }_v^2}}{\displaystyle \sum _{j=1}^{N_v}}{\left({\displaystyle \frac{\partial u^{\prime }(s_j;{\mu }^{*})}{\partial \mu }}\right)}^2.$$

Since all terms are nonnegative, ${\mathcal{I}}_D\left({\mu }^{*}\right)\ge {\mathcal{I}}_P\left({\mu }^{*}\right)$. The inequality is strict whenever at least one derivative observation has nonzero velocity sensitivity $\partial u^{\prime }(s_j;{\mu }^{*})/\partial \mu \ne 0$. □

Interpretation. Theorem 2 provides an information-theoretic justification for D-PINN: derivative observations add a nonnegative term to the Fisher information, which (by the Cramér–Rao bound) reduces the minimum achievable variance of any unbiased estimator of $\mu$. The gain is proportional to the sensitivity of the velocity to $\mu$, which is typically large for damping parameters (since $\mu$ multiplies $u^{\prime }$ in the governing equation). In regions where $u^{\prime }\left(t\right)\approx 0$ (e.g., at displacement extrema), derivative observations provide less information, explaining why a sufficient density of derivative observation points is beneficial.

5.3. Residual-Based Parameter Stability Estimate

Theorem 3.

Let $u^{*}\left(t\right)$ be the true solution satisfying $m{u^{*}}^{\prime \prime }+{\mu }^{*}{u^{*}}^{\prime }+k{u}^{*}=f$, and let $u_{\theta }\left(t\right)$ be a neural network approximation with residual $R_{\theta }\left(t\right)=m{u}_{\theta }^{\prime \prime }+{\mu }_{\theta }{u}_{\theta }^{\prime }+k{u}_{\theta }-f$. Define $e=u_{\theta }-u^{*}$. Then the parameter error satisfies

$$|{\mu }_{\theta }-{\mu }^{*}|\le {\displaystyle \frac{\parallel R_{\theta }{\parallel }_{L^2}+m\parallel e^{\prime \prime }{\parallel }_{L^2}+|{\mu }_{\theta }|\parallel e^{\prime }{\parallel }_{L^2}+k{\parallel e\parallel }_{L^2}}{\parallel {u^{*}}^{\prime }{\parallel }_{L^2}}},$$

(17)

provided $\parallel {u^{*}}^{\prime }{\parallel }_{L^2}>0$ (the true solution has nonzero velocity in the $L^2$ sense).

Proof. 

Subtract the true governing equation from the residual definition:

$$\begin{array}{cc}\hfill R_{\theta }& =(m{u}_{\theta }^{\prime \prime }+{\mu }_{\theta }{u}_{\theta }^{\prime }+k{u}_{\theta }-f)-(m{u^{*}}^{\prime \prime }+{\mu }^{*}{u^{*}}^{\prime }+k{u}^{*}-f)\hfill \\ \hfill & =m{e}^{\prime \prime }+{\mu }_{\theta }{e}^{\prime }+({\mu }_{\theta }-{\mu }^{*}){u^{*}}^{\prime }+ke.\hfill \end{array}$$

Rearranging to isolate the parameter error term:

$$({\mu }_{\theta }-{\mu }^{*}){u^{*}}^{\prime }=R_{\theta }-m{e}^{\prime \prime }-{\mu }_{\theta }{e}^{\prime }-ke.$$

Taking $L^2$ norms and applying the triangle inequality:

$$|{\mu }_{\theta }-{\mu }^{*}|·\parallel {u^{*}}^{\prime }{\parallel }_{L^2}\le \parallel R_{\theta }{\parallel }_{L^2}+m\parallel e^{\prime \prime }{\parallel }_{L^2}+|{\mu }_{\theta }|\parallel e^{\prime }{\parallel }_{L^2}+k{\parallel e\parallel }_{L^2}.$$

Dividing by $\parallel {u^{*}}^{\prime }{\parallel }_{L^2}$ (which is positive for any non-trivial solution) yields (17). □

Interpretation. Theorem 3 decomposes the parameter error into four components: the physical residual $\parallel R_{\theta }\parallel$, the second-derivative error $\parallel e^{\prime \prime }\parallel$, the first-derivative error $\parallel e^{\prime }\parallel$, and the displacement error $\parallel e\parallel$. Standard PINNs primarily minimize $\parallel R_{\theta }\parallel$ and $\parallel e\parallel$, leaving $\parallel e^{\prime }\parallel$ and $\parallel e^{\prime \prime }\parallel$ weakly constrained. D-PINN directly targets $\parallel e^{\prime }\parallel$ through the derivative observation loss, which reduces the $|{\mu }_{\theta }|\parallel e^{\prime }\parallel$ term in the bound. Combined with Theorem 1 (which controls $\parallel e\parallel$ via $\parallel e^{\prime }\parallel$), this provides a chain of error control: derivative supervision → smaller $\parallel e^{\prime }\parallel$→ smaller $\parallel e\parallel$→ tighter parameter bound.

5.4. Interpretation and Limitations of the Theory

The three theorems provide complementary perspectives on why derivative observations improve parameter identifiability, but they do not guarantee global convergence of the training optimization. Theorem 1 is a deterministic functional inequality that holds for any $C^1$ function vanishing at the origin. Theorem 2 is a local (asymptotic) information-theoretic result valid in a neighborhood of ${\mu }^{*}$. Theorem 3 is a deterministic error bound that is valid for any network output but does not account for the optimization dynamics that determine whether the right-hand side actually becomes small. In practice, the optimization landscape of PINNs is non-convex, and gradient-based training may converge to local minima. The theoretical results should be understood as explaining why derivative observations are beneficial when training succeeds, not as a proof that training always succeeds.

6. Numerical Experiments

6.1. Experimental Setup

All experiments use the following standardized configuration unless otherwise specified. System parameters: $m=1.0$, $k=400.0$, ${\mu }^{*}=4.0$, ${\omega }_0=20$, $\delta =2$, time domain $t\in [0,1]$. Observation settings: $N_u=20$ displacement points, $N_v=20$ derivative points, $N_c=30$ collocation points, all uniformly spaced. Training: 20,000 epochs with Adam optimizer (initial learning rate ${10}^{-3}$, cosine annealing to ${10}^{-5}$). Loss weights: ${\lambda }_{\mathrm{phys}}=5$, ${\lambda }_{\mathrm{data}}=10$, ${\lambda }_{\mathrm{deri}}=0.1$, ${\lambda }_{\mathrm{init}}=1$, ${\lambda }_{\mathrm{reg}}=0.01$ when prior is used. Network: 4 hidden layers × 128 neurons, tanh activation, Fourier features with 6 frequencies $f_i=2^{i-1}$. Each method is run with 10 independent random seeds (varying network initialization, observation sampling, and noise realization), and results are reported as mean ± standard deviation.

6.2. Main Results on the Linear Oscillator

Table 1 presents the comprehensive method comparison on the homogeneous linear underdamped oscillator.

Several findings merit detailed discussion:

Classical NLS achieves near-perfect parameter recovery ($\mu$ error $<{10}^{-7}$, relative error $\approx 0\%$). Under the current synthetic setting where the system model is exactly correct and observations are noise-free, NLS using the analytical solution converges to machine precision. This establishes an important baseline: when the forward model is exact, classical optimization remains highly competitive. The value of D-PINN lies in scenarios where the forward model is uncertain or where differentiability with respect to parameters is difficult to obtain.

Standard PINN, Fourier-PINN, and gPINN exhibit large parameter errors ($40.0\%$, $69.7\%$, and $40.1\%$ relative error, respectively). Notably, Fourier-PINN performs worse than standard PINN in parameter estimation, despite achieving better displacement fitting (MSE $4.49\times {10}^{-2}$ vs. $1.29\times {10}^{-1}$). This result shows that improved function approximation does not necessarily lead to more accurate parameter identification, highlighting the need to evaluate these two objectives separately in PINN benchmark studies. The gPINN residual gradient penalty provides no measurable benefit for this inverse problem, consistent with the interpretation that residual smoothness does not add information about $\mu$.

Sobolev-PINN achieves the smallest displacement error but completely fails at parameter identification (MSE $3.26\times {10}^{-6}$, $\mu$ error $1.5000\pm 0.0000$, $\mu$ rel. error $37.5\%$). The $\mu$ estimate is stuck at exactly $2.5$, the initial value. This demonstrates a critical insight: accurate function approximation (even including derivative matching) without physics residual constraints provides no driving force for parameter learning. The physics residual is essential for $\mu$ to move away from its initialization.

D-PINN without prior reduces $\mu$ error by a factor of 3.4 compared to standard PINN ($0.467\pm 0.259$ vs. $1.600\pm 0.069$). This improvement is attributable solely to the derivative observation loss, as no parameter regularization is used. The displacement MSE also improves from $1.29\times {10}^{-1}$ to $4.38\times {10}^{-4}$.

D-PINN without Fourier features reverts to near-standard-PINN performance ($\mu$ error $1.571\pm 0.058$). This ablation confirms that the Fourier feature mapping is essential for D-PINN to effectively utilize derivative observations; without it, the network cannot adequately represent the oscillatory velocity signal.

D-PINN with weak prior${\mu }_0=3.2$achieves the best PINN-family result ($\mu$ error $0.084\pm 0.051$, relative error $2.10\%\pm 1.27\%$). This is a 19-fold improvement over standard PINN and a 5.6-fold improvement over D-PINN without prior. Notably, ${\mu }_0=3.2$ differs from the true value by $20\%$ and therefore does not represent an oracle setting. Nevertheless, the result shows that combining derivative observations with a coarse prior can substantially improve parameter estimation. The regularization weight ${\lambda }_{\mathrm{reg}}=0.01$ is small enough that the prior does not dominate the loss.

Prior quality matters. D-PINN with weak prior ${\mu }_0=4.8$ ($20\%$ overestimate) achieves $\mu$ error $0.981\pm 0.107$ ($24.5\%$ relative error), which is worse than no prior. D-PINN with wrong prior ${\mu }_0=5.5$ ($37.5\%$ overestimate) achieves $\mu$ error $1.427\pm 0.129$ ($35.7\%$), nearly as poor as standard PINN. The underestimated prior (${\mu }_0=3.2$) yields a greater improvement than the overestimated prior (${\mu }_0=4.8$). One possible explanation is that lower values of $\mu$ produce slower decay and may increase the sensitivity of the displacement response to parameter variations in this direction.

Oracle prior (${\mu }_0=4.0$) gives $\mu$ error $0.497\pm 0.086$, which is worse than the weak prior ${\mu }_0=3.2$ result. This counterintuitive outcome suggests that a prior exactly at the true value does not guarantee optimal performance when combined with derivative observations and physics residuals, possibly due to competing gradient signals in the composite loss. We emphasize that the oracle prior is reported for diagnostic completeness only and should not be interpreted as a practical result.

6.3. Forced Vibration and Duffing Oscillator

Figure 1 and Figure 2 show representative displacement fitting results for the forced vibration (near-resonance, $F_0=20$, ${\omega }_f=19$) and Duffing oscillator ($\alpha =20$) systems. Detailed tabular results from the 10-seed experiments are summarized below.

Forced vibration. NLS achieves $\mu$ error $0.0037\pm 0.0000$. Standard PINN exhibits catastrophic behavior: $\mu$ error $35.38\pm 21.99$, with individual seeds diverging to $\mu >80$. This extreme variance indicates that the near-resonance forcing creates a highly ill-conditioned inverse problem for displacement-only PINNs. D-PINN without prior achieves $\mu$ error $0.727\pm 0.361$, and D-PINN with weak prior ${\mu }_0=3.2$ achieves $0.666\pm 0.308$. While the absolute errors are larger than in the homogeneous case, D-PINN provides critical stabilization, preventing the catastrophic divergence observed with standard PINN.

Duffing oscillator. NLS achieves $\mu$ error $0.0040\pm 0.0000$. Standard PINN achieves $\mu$ error $1.597\pm 0.074$ ($39.9\%$ relative error), consistent with the homogeneous linear case. D-PINN without prior achieves $\mu$ error $0.382\pm 0.212$ ($9.6\%$), a 4.2-fold improvement. D-PINN with weak prior ${\mu }_0=3.2$ achieves $\mu$ error $0.080\pm 0.081$ ($2.0\%$), demonstrating that the combination of derivative observations and a coarse prior generalizes to nonlinear systems.

6.4. Prior Sensitivity Analysis

Table 2 quantifies the sensitivity of D-PINN’s $\mu$ estimation to the prior value. The experiment uses the linear homogeneous oscillator with identical training configuration, varying only ${\mu }_{\mathrm{prior}}$ and ${\lambda }_{\mathrm{reg}}$.

The prior sensitivity analysis, shown in Figure 3 and Table 2, reveals an asymmetric response: underestimating priors (${\mu }_0=2.5,3.2$) produce lower $\mu$ error than overestimating priors (${\mu }_0=4.8,5.5$) of equal percentage deviation. This asymmetry is consistent with the physics: a lower damping coefficient produces a more slowly decaying displacement envelope, making the response more sensitive to $\mu$ in that direction. The wrong prior ${\mu }_0=5.5$ ($37.5\%$ deviation) yields $\mu$ error $1.427\pm 0.129$, which is worse than using no prior at all ($0.467\pm 0.259$), underscoring the risk of overconfident or incorrect priors.

6.5. Derivative Observation Source Analysis

In practical applications, velocity measurements may be obtained through various means with different accuracy and noise characteristics. Table 3 compares four derivative observation sources under identical training conditions (linear oscillator, ${\sigma }_u=0.02$ displacement noise, D-PINN without prior).

Figure 4. Effect of derivative observation source on $\mu$ estimation accuracy. Exact and noisy velocity perform similarly (both within measurement uncertainty). Finite-difference derivatives degrade accuracy by approximately $60\%$, and Savitzky–Golay smoothed derivatives degrade by approximately $270\%$.

Figure 4. Effect of derivative observation source on $\mu$ estimation accuracy. Exact and noisy velocity perform similarly (both within measurement uncertainty). Finite-difference derivatives degrade accuracy by approximately $60\%$, and Savitzky–Golay smoothed derivatives degrade by approximately $270\%$.

Exact and noisy velocity observations produce nearly identical results ($\mu$ error $0.430\pm 0.214$ vs. $0.449\pm 0.248$), indicating that moderate Gaussian noise (${\sigma }_v={\sigma }_u=0.02$) in the velocity measurements does not significantly degrade D-PINN performance. Finite-difference derivatives computed from noisy displacement data increase $\mu$ error to $0.689\pm 0.272$ ($+60\%$), and Savitzky–Golay smoothed finite differences further degrade performance to $1.586\pm 0.356$ ($+269\%$). The SG filter, while reducing high-frequency noise in the displacement signal, appears to oversmooth the derivative estimate, removing genuine velocity variation that carries information about $\mu$. These results suggest that direct velocity measurements (via laser Doppler vibrometry or accelerometer integration) are preferable when available, but finite-difference derivatives remain a viable alternative when only displacement sensors are available.

6.6. Discussion of Incomplete or Failed Experiments

The Wavelet PINN baseline from the original conference manuscript was excluded from this study because preliminary experiments showed inconsistent convergence behavior across seeds. The wavelet activation introduces additional trainable parameters (scale and shift) that interact unpredictably with the derivative observation loss, making fair comparison difficult. This baseline is superseded by the more principled gPINN and Sobolev-PINN baselines.

Bayesian uncertainty quantification (e.g., via Hamiltonian Monte Carlo or B-PINN [4]) is not implemented in the current study. While uncertainty estimates for $\widehat{\mu }$ would strengthen the identifiability analysis, they require substantially more computation and are left for future work.

7. Discussion

7.1. When Derivative Observations Help

The experimental results support a nuanced picture of when derivative observations improve parameter estimation:

1.

Linear homogeneous systems with sparse displacement data. Derivative observations provide the largest benefit, reducing $\mu$ error by a factor of 3.4 without any prior. This is the regime where the parameter degeneracy identified in Section 3.3 is most severe.

2.

Forced/nonlinear systems. Derivative observations provide important stabilization by preventing the catastrophic divergence observed with the standard PINN in the forced-vibration case. However, the absolute parameter errors remain larger than those in the homogeneous case.

3.

Combined with a weak prior. The combination of derivative observations and a coarse prior (${\mu }_0=3.2$, $20\%$ deviation) achieves the best results across all three systems, reducing $\mu$ relative error to $2.1\%$ (linear), improving forced vibration stability, and achieving $2.0\%$ on the Duffing oscillator.

7.2. When Derivative Observations Do Not Help

1.

Displacement fitting. Standard PINN already achieves displacement MSE of $1.29\times {10}^{-1}$ on the linear oscillator; D-PINN improves this only marginally in some cases. D-PINN should not be presented as a general-purpose accuracy improvement—its benefit is specifically in parameter estimation.

2.

With poor-quality derivative estimates. When derivatives are obtained through Savitzky–Golay smoothing of noisy displacement data, D-PINN performance degrades below the no-derivative baseline.

3.

Without Fourier features. D-PINN without Fourier features ($\mu$ error $1.571$) performs nearly identically to standard PINN ($\mu$ error $1.600$), indicating that the network must be capable of representing the oscillatory velocity signal for derivative supervision to be effective.

7.3. Limitations

1.

Single-degree-of-freedom systems. The current study is limited to scalar second-order ODEs. Extension to multi-degree-of-freedom systems and partial differential equations requires further investigation.

2.

Synthetic data. All experiments use synthetic data generated from known ODE solutions. Real experimental data would introduce model-form uncertainty (the governing equation may not be exact) and sensor-specific noise characteristics.

3.

Hyperparameter sensitivity. The loss weights were tuned on the linear oscillator and applied unchanged to forced and Duffing systems. Adaptive loss balancing methods [12] may improve robustness across systems.

4.

Computational cost. NLS achieves near-perfect results on these synthetic benchmarks with orders of magnitude less computation than D-PINN. The value proposition of D-PINN lies in scenarios where the forward model is uncertain, differentiability with respect to parameters is difficult to obtain, or the system is not expressible in closed form—scenarios not captured by the current synthetic benchmarks.

8. Conclusions

This paper proposed D-PINN, a derivative-observation-augmented physics-informed neural network for parameter identification in second-order dynamical systems, and provided a theoretical analysis of why derivative observations improve parameter identifiability. Three theoretical results, comprising a Sobolev-type error bound, a Fisher information decomposition, and a residual-based stability estimate, show that derivative supervision controls displacement error, contributes nonnegative information, and yields a tighter bound on the parameter estimation error.

Numerical experiments with 10 random seeds and 20,000 training epochs on three dynamical systems support the following conclusions:

1.

D-PINN without any parameter prior reduces the damping coefficient relative error from $40.0\%$ (standard PINN) to $11.7\%$, a 3.4-fold improvement attributable solely to derivative observations.

2.

With a weak, non-oracle prior (${\mu }_0=3.2$ vs. true ${\mu }^{*}=4.0$), D-PINN achieves $2.1\%$ relative error, a 19-fold improvement over standard PINN, while maintaining a prior $20\%$ away from the true value.

3.

The quality of derivative observations matters: finite-difference derivatives degrade performance by $60\%$, and Savitzky–Golay smoothed derivatives degrade by $270\%$ relative to exact velocity measurements.

4.

D-PINN provides critical stabilization for forced vibration near resonance, where standard PINN exhibits catastrophic divergence (some seeds exceeding $2000\%$ relative error).

5.

D-PINN does not improve displacement fitting accuracy in settings where standard PINN already achieves low MSE; its benefit is specifically in parameter estimation.

Future work includes extending the derivative observation framework to multi-degree-of-freedom systems, incorporating adaptive loss balancing for cross-system robustness, and validating D-PINN on experimental vibration data with real sensor noise and model-form uncertainty.