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A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response
Yi An, Zhijiang Chen, Zhiqiang Feng, Qian Cheng, Jack C. P. Cheng, Haijiang Li, Dalei Wang
TL;DR
PINNs can struggle to converge to higher-order eigenmodes and model multi-frequency structural dynamics. The paper integrates dynamic stiffness matrices with PINNs, using the Wittrick-Williams algorithm and frequency-domain modeling, and reports accurate arbitrary-order modes with strong FEM agreement and efficient convergence.
Problem
PINNs may converge to lower-order modes and struggle to represent the multi-frequency functions common in structural dynamics.
Method
DSM-PINNs integrate dynamic stiffness matrices with PINNs, using the Wittrick-Williams algorithm for eigenvalues and frequency-domain modeling for dynamic responses.
Results
The method determines arbitrary-order eigenmodes with strong FEM agreement and efficient convergence, while dynamic-response errors reach a maximum of approximately 0.027.
Takeaways & Limitations
DSM-PINNs provide a practical framework for structural eigenvalue and dynamic-response analysis within the studied cases.
Takeaways & Limitations
The motivating neural-network limitations include failure to converge to high-order modes and difficulty modeling multi-frequency functions.
Abstract
from arXiv · showhide
This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.
1 Introduction
The paper motivates a DSM-PINN framework by addressing neural-network spectral bias and limitations in structural eigenvalue and dynamic-response analysis. It embeds the dynamic stiffness method and Wittrick–Williams algorithm into physics-informed neural networks for robust frequency-domain analysis.
- Motivation: Structural free-vibration and dynamic-response analysis is important for structural design optimization and safety across civil engineering, aerospace, automotive, and shipbuilding.
- Motivation: Physics-informed deep learning incorporates observational information, physical rules, and empirical findings to address shortcomings of exclusively data-driven models.
- Limitations: Neural networks may converge to lower-order rather than high-order feature modes because of insufficient similarity with high-order feature functions, reflecting spectral bias.
- Proposed framework: The proposed framework combines the Dynamic Stiffness Matrix with Physics-Informed Neural Networks and exploits DSM advantages over FEM’s approximate interpolation functions and energy variational principles.
- Contributions: The eigenmode of an arbitrary-order structure is reformulated as a neural-network lowest-eigenvalue problem using the Wittrick–Williams algorithm.
- Contributions: A robust frequency-domain physics-informed neural-network framework combines experimental data with physical laws.
2 The physics-informed neural networks and their application in finite element method
This section presents FEM-integrated PINNs that approximate nodal displacements with neural networks and embed stiffness-matrix physics into energy-based or equilibrium-based loss functions. It also describes boundary-condition enforcement, optimization, and local convergence stability.
- PINN formulation: FEM-integrated PINNs approximate nodal solution variables with neural networks over discrete elements and optimize parameters using potential-energy or equilibrium-based physical losses.The global displacement vector is represented by N(x; θ), while K and P define the stiffness and load terms.
- Optimization: Neural-network parameters are optimized with first-order or quasi-Newton methods, including SGD, Adam, and L-BFGS, using gradients of the physical or complete loss.The update uses the learning rate γ_n and the first-order gradient with respect to θ.
- Boundary conditions: Boundary conditions are enforced as natural conditions through nodal loads, soft conditions through loss data, or hard conditions through customized network outputs or stiffness-matrix modification.Penalty enforcement multiplies the relevant diagonal stiffness term by β and replaces the corresponding load entry for nonzero prescribed displacement.
- Convergence stability: Energy-based FEM PINNs exhibit local stability because the stiffness matrix is positive semi-definite and the constrained matrix K∗ is positive.At convergence, the loss gradient and Hessian-related derivative terms tend toward zero.
- Convergence stability: Nodal-equilibrium FEM PINNs likewise exhibit local stability because (K∗)^2 remains positive semi-definite or positive under sufficient constraints.The equilibrium-loss gradient vanishes as the neural network reaches an extremum, while the corresponding second-derivative terms remain zero at convergence.
3 Addressing the eigenvalue problems in the dynamic stiffness matrix using PINNs
This section develops a DSM-PINN eigenvalue procedure in which exact dynamic stiffness matrices provide a strong-form structural solution, while the Wittrick-Williams algorithm transforms arbitrary-order eigenvalue searches into lowest-order generalized eigenproblems. Normalized neural-network eigenvectors and inverse iteration address nontrivial-mode solutions and low-order convergence, with numerical results agreeing closely with FEM and prior studies.
- DSM formulation: Exact homogeneous-solution shape functions produce the dynamic stiffness matrix, which combines with PINN nodal equilibrium equations to yield a strong-form structural solution.The Euler-Bernoulli beam formulation derives a 4×4 DSM from exact matrices R and U, avoiding energy variation and targeting analytical convergence.
- Eigenvalue transformation: The Wittrick-Williams algorithm transforms the arbitrary j-th order eigenvalue problem into a lowest-order generalized eigenproblem within a small frequency interval.The method uses lower and upper circular-frequency bounds and a Taylor expansion about an approximate eigenvalue; the interval contains one eigenvalue.
- PINN challenges: PINNs face nontrivial-solution normalization and low-order-mode convergence challenges when solving homogeneous eigenvalue equations.The eigenvalue formulation must obtain and normalize nontrivial eigenmodes, while training tends to converge toward low-order eigenmodes.
- PINN solution procedure: Normalizing the eigenvector at every iteration and transforming arbitrary modes into lowest-order problems resolves the two principal neural-network difficulties.Inverse-iteration-inspired training approximates the eigenvector, while the transformed problem determines the smallest generalized eigenvalue and associated normalized mode.
- Numerical validation: The proposed method agrees closely with previous studies and FEM analyses, including the first three eigenfrequencies and corresponding modeshapes.Reported comparisons also show satisfactory concordance at an identical iteration tolerance of 1e−4, while modeshape discrepancies occur for modes 3 and 5 in the cantilever case.
- Convergence behavior: Higher-order eigenvalues require more training iterations because their eigenvectors are more tortuous, although convergence efficiency remains high.The loss-function evolution attributes the increased iteration count to greater spatial and temporal variation associated with higher-order deformation waves.
4 Addressing the structural dynamics using a frequency domain PINNs
The frequency-domain DSM-PINN converts structural dynamics into a frequency-domain problem, using separate real- and imaginary-part networks and inverse Fourier transformation to recover time-domain responses. Its results show low errors against FEM and efficient convergence, while requiring a loss formulation compatible with the DSM’s indefinite character.
- Complex-valued response: Two neural networks separately represent the real and imaginary displacement components because neural networks cannot directly process complex numbers.The DSM-PINN approximates both displacement and rotation in their real and imaginary parts.
- Frequency-domain formulation: The FD-PINN applies a Fourier transform to obtain a dynamic stiffness matrix and frequency-domain load, then solves for the response spectrum.The discrete inverse Fourier transform recovers the governing-equation solution in the time domain.
- Dynamic stiffness formulation: DSM shape functions derive from homogeneous solutions rather than polynomials, so equivalent nodal forces are exact.The global nodal force vector is assembled from these elemental forces, producing equations that represent the original PDEs.
- Convergence stability: The energy-type loss is unsuitable for DSMs, whereas the nodal-equilibrium loss can be used when the Hessian is positive definite.The DSM becomes indefinite depending on trial frequency, requiring a compatible strong-form PINN loss.
- Numerical results: Maximum relative error was approximately 0.027 across three moving-load cases compared with FEM, with convergence typically within 100 iterations.The calculated natural frequencies were ω1 = 22.112 rad/s and ω2 = 88.275 rad/s.
- Numerical results: Maximum relative error was almost 0.0375 across three cases compared with FEM, while convergence typically required fewer than 200 iterations.The real part at ω = 0 rad/s was an exception, and frequencies near the first-order mode or the fourth-order imaginary response required more iterations.
5 Conclusions
DSM-PINNs provide a flexible framework for solving homogeneous vibration eigenvalue problems and nonhomogeneous dynamic responses by integrating neural networks with the dynamic stiffness matrix. The conclusions report successful eigenvalue tracing, efficient convergence, frequency-domain response analysis, and extensions beyond beam structures.
- Framework: DSM-PINNs solve homogeneous eigenvalue and nonhomogeneous dynamic response problems through a framework centered on the dynamic stiffness matrix W.Although demonstrated for Euler–Bernoulli beams, the core step is associated with W and is not limited to beams.
- Eigenvalue problems: The Wittrick-Williams algorithm addresses nontrivial-solution acquisition and neural-network convergence toward lowest-order eigenmodes by transforming arbitrary-order eigenvalues into lowest-order form.The algorithm restricts eigenvalues within narrow bounds.
- Dynamic response: FD-PINNs use the DSM as a frequency-domain spectral element to address spectral bias in multi-frequency functions and reduce neural-network modeling complexity.Transforming displacement fields and loads into the frequency domain incorporates differential terms into the DSM.
- Extensions: The method can extend to plates and shells when DSM W is determined and is compatible with standard FEM-PINNs and deep operator neural networks.The compatibility follows similarities between DSM and FEM treatments of eigenvalue problems and dynamic responses.
Appendix
The appendix gives explicit Euler–Bernoulli beam dynamic-stiffness expressions, their Taylor-series expansion, frequency-count definitions, frame-matrix augmentation, and moving-load force formulation.
- Beam dynamic stiffness matrix: The Euler–Bernoulli beam DSM is explicitly represented using trigonometric and hyperbolic functions from the homogeneous beam solution.The appendix defines c = cos α, s = sin α, C = cosh α, and S = sinh α.
- Matrix expansion: The DSM parameters δ, δl, θ, θl, β, and γ are defined in terms of α, D, and trigonometric-hyperbolic combinations, and the matrix can be expanded by a first-order Taylor series.The expansion uses FEM stiffness and mass matrices K and M.
- Frequency counting: The appendix defines j0 and jm as natural-frequency counts under fully restrained and fully fixed boundary-condition assumptions, respectively.j0 counts frequencies between ω = 0 and ω = ω*, while jm counts frequencies associated with each individual element.
- Frame and moving-load formulation: For two-dimensional frames, the stiffness matrix is augmented with an axial dynamic-stiffness component, while moving-load forces are represented by a 4 × 1 nodal-force vector.The axial term is associated with the longitudinal wave phase speed, and fd denotes the moving point-load force vector.