PSI - Issue 84

Gianluca Bruno et al. / Procedia Structural Integrity 84 (2026) 240–247

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The logic of the agent is to maximize the expected cumulative return within the episode. To address the issues related to high-dimensional action spaces (as in the case of structural model updating), Actor-Critic architectures are preferred. In this work, the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm is adopted (Fujimoto et al. 2018). In the proposed context, the agent learns to tune the uncertain parameters of the numerical model by progressively observing the reduction in error between the simulated (by model) and experimental (by data) dynamic responses.

Fig. 1. Interaction between agent and environment in DRL.

2.2. Fundamentals of reduced-order modelling A ROM is a mathematical approximation to a high-fidelity model, created to replicate the essential dynamic behavior of a structure. The primary goal of model reduction is to preserve dominant physical characteristics of the system, while reducing degrees of freedom with respect to FOM. The transition from the FOM to ROM takes its starting point from the equation of motion, which characterizes the dynamic problem, and that can be defined in its compact form as ̈( ) + ̇( ) + ( ) = ( ) (1) where , , ∈ ℝ × are the mass, damping, and stiffness matrices of the FOM, respectively; ( ), ̇( ), ̈( ) ∈ ℝ are the vectors of nodal displacements, pseudo-velocities, and pseudo-accelerations, respectively; ( ) represents the vector of external forces. The response is approximated by projecting the state vector onto a reduced subspace of dimension ≪ using a transformation matrix Φ∈ℝ × such that ( ) ≈ Φ ( ) , where ( ) ∈ℝ is the vector of reduced or generalized coordinates (Benner et al. 2015). By applying Galerkin projection to Eq. 1, the reduced system can be expressed as ̈( ) + ̇( ) + ( ) = ( ) (2) which can be re-edited as ̈ ( ) + ̇ ( ) + ( ) = ( ) (3) where , , ∈ℝ × are the reduced order matrices and ∈ℝ is the vector of reduced order forces, presenting significantly smaller size than the ones in Eq. 1 and preserve the overall dynamic properties of the original system. The definition of the reduction basis Φ is crucial for the accuracy of the ROM, and to this scope the Proper Orthogonal Decomposition (POD) technique is used. POD is a powerful data-driven technique for extracting an optimal orthonormal basis from a set of experimental or numerical data (Vlachas et al. 2015). For the case at hand, a collection of a snapshot matrix ∈ℝ × is performed, obtained by assembling the system responses (e.g., displacement vectors or modal shapes) computed by the FOM for points in the time or for different combinations h of the characteristic parameter values, with ℎ = 1,…, .. The solutions (ℎ) are assembled as columns of a matrix :