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Physics-informed neural networks for solving forward and inverse problems in complex beam systems

Taniya Kapoor, Hongrui Wang, Alfredo Nunez, Rolf Dollevoet

arXiv:2303.01055v2cs.LGcs.NAmath.NA

TL;DR

The paper addresses forward and inverse problems for complex beam systems, where PDE solutions can be difficult to obtain and inverse inputs may be unknown. It applies PINNs with nondimensional equations to beam models, reporting accurate forward predictions and inverse identification, including under noisy data.

  • Problem

    Numerical PDE methods can face mesh-creation difficulties in complex geometries, and inverse beam problems may have unknown model inputs such as coefficients or applied forces.

    Source: §I · pp. 1–2

  • Method

    The paper uses PINNs with nondimensional PDEs to solve forward and inverse problems for Euler–Bernoulli and Timoshenko beam systems.

    Source: §I · p. 2

  • Results

    For the double Timoshenko beam, displacement prediction had less than 5e −3% error with 1600 training points, and force identification had less than 0.2% error with 20% Gaussian noise.

    Source: §V · p. 13

  • Takeaways & Limitations

    The reported nonincrease in error as model complexity increases suggests the framework is appropriate for larger systems with multiple connected components.

    Source: §V · p. 13

Abstract

from arXiv · show

This paper proposes a new framework using physics-informed neural networks (PINNs) to simulate complex structural systems that consist of single and double beams based on Euler-Bernoulli and Timoshenko theory, where the double beams are connected with a Winkler foundation. In particular, forward and inverse problems for the Euler-Bernoulli and Timoshenko partial differential equations (PDEs) are solved using nondimensional equations with the physics-informed loss function. Higher-order complex beam PDEs are efficiently solved for forward problems to compute the transverse displacements and cross-sectional rotations with less than 1e-3 percent error. Furthermore, inverse problems are robustly solved to determine the unknown dimensionless model parameters and applied force in the entire space-time domain, even in the case of noisy data. The results suggest that PINNs are a promising strategy for solving problems in engineering structures and machines involving beam systems.

I. INTRODUCTION

Complex beam systems pose challenging PDE problems, and dimensional equations can defeat PINN training when their coefficients are large. The paper motivates nondimensional PINNs for forward and inverse beam problems and demonstrates the dimensional Euler–Bernoulli difficulty.

I. INTRODUCTION

  • Complex engineering systems are difficult to analyze conventionally, while numerical PDE methods can face mesh-creation challenges in complex, higher-dimensional geometries.PDE solutions support structural-health monitoring and optimal structural design.Source: §I · p. 1
  • The paper proposes nondimensional PINNs for forward and inverse Euler–Bernoulli and Timoshenko beam problems, identifying unknown parameters and forces from indirect measurements.The stated measurements include displacement and cross-sectional rotations.Source: §I · p. 2

II. PINNS FOR DIMENSIONAL PDES

  • The dimensional Euler–Bernoulli forward problem computes transverse displacement from the governing PDE, initial and boundary conditions, and applied force.The described test uses a uniform cross-section beam with constant material properties and simply supported ends.Source: §II · p. 3
  • PINNs embed the physical PDE in a loss function alongside initial and boundary conditions; forward problems require no additional data in that loss.The neural network maps space and time inputs to the quantity of interest.Source: §II · p. 3
  • Large PDE coefficients make the dimensional Euler–Bernoulli PINN fail to converge, motivating nondimensionalization for computational tractability.After 15,000 epochs, the optimizer remains at a vast training loss and is stuck in a local minimum.Source: §II · pp. 3–4

III. PINNS FOR NONDIMENSIONAL PDES

The section transforms dimensional variables into nondimensional ones and uses the resulting PDE in the PINN framework. It presents nondimensionalization as a way to improve computational stability, generalization, and interpretability.

III. PINNS FOR NONDIMENSIONAL PDES

  • The framework maps dimensional space, time, displacement, and force quantities to corresponding nondimensional quantities before rewriting the governing PDE.The transformed equation uses a dimensionless model parameter.Source: §III · p. 4
  • Nondimensional equations keep variables and parameters within a specific range, which the paper says improves neural-network performance and generalization.The framework also presents dimensionless equations as improving computational stability and interpretability.Source: §III · p. 4
  • Removing units makes solutions easier to interpret and enables comparisons across physical systems using ratios and parameters.Source: §III · p. 4

A. PINN Framework for Forward Problems

The forward-problem framework places the nondimensional PDE in the PINN loss function and is illustrated schematically.

A. PINN Framework for Forward Problems

B. Nondimensional Euler-Bernoulli Beam Equation

The section nondimensionalizes the Euler-Bernoulli beam equation and evaluates a PINN solution against an analytical solution. The predicted displacement has a relative percentage error of 5.3e-4 percent, with higher errors where training points are sparse.

B. Nondimensional Euler-Bernoulli Beam Equation

  • Nondimensionalization yields utt + uxxxx = f(x, t) on x ∈ [0, π] and t ∈ [0, 1], with the stated sinusoidal force and simply supported boundary conditions.The variables are scaled as u = ū/l, x = x̄/l, t = c t̄/l², and f = f̄l³/(EI).Source: B. Nondimensional Euler-Bernoulli Beam… · p. 4
  • The error measure R is a relative percentage error comparing the network prediction with the analytical solution.The analytical solution used in this case is u(x, t) = sin(x) cos(4πt).Source: B. Nondimensional Euler-Bernoulli Beam… · p. 4
  • 5.3e-4 percent relative percentage error is reported for the nondimensional Euler-Bernoulli beam displacement prediction.Fig. 3 presents the predicted solution and its absolute error.Source: B. Nondimensional Euler-Bernoulli Beam… · p. 4, Figure 3 · p. 5
  • Higher errors occur in regions with fewer training points, while the overall error remains low and the network captures displacement behavior accurately.The paper notes that adding training points in high-error regions is one approach to reduce error.Source: B. Nondimensional Euler-Bernoulli Beam… · p. 4

C. PINN Framework for Inverse Problems

The paper treats inverse problems as ill-posed because parameters, forces, or initial and boundary conditions may be unknown. Its PINN approach uses observed solution data to estimate unknown parameters or forces while modifying the loss function.

C. PINN Framework for Inverse Problems

  • Inverse problems are ill-posed when initial or boundary conditions, parameters, or forces are unknown.The corresponding abstract dimensionless PDE is contrasted with a well-posed forward problem that has a unique solution.Source: C. PINN Framework for Inverse Problems · p. 5
  • Given observable u data from part of the training domain, the inverse problem aims to predict an unknown parameter λ or force function f(x, t).The available inverse-problem observations are denoted udata at Ndata points.Source: C. PINN Framework for Inverse Problems · p. 5
  • The inverse PINN modifies the forward-problem loss by adding data and can also predict unknown forces, parameters, or initial and boundary conditions.The framework schematic distinguishes forward losses using PDEs and initial and boundary conditions from inverse losses supplemented by data.Source: C. PINN Framework for Inverse Problems · p. 5, Figure 5 · p. 6

IV. NUMERICAL EXPERIMENTS AND DISCUSSION

The numerical experiments progress from single-beam systems to more complex double beams connected to a Winkler foundation. They use single-beam cases as proof of concept before applying the method to double-beam forward and inverse problems.

IV. NUMERICAL EXPERIMENTS AND DISCUSSION

  • The experiments progress from simple single-beam models to double beams connected to a Winkler foundation.Source: §IV · p. 6
  • Single-beam forward and inverse problems serve as the proof of concept for the proposed method.Source: §IV · p. 6
  • The method is then applied to forward and inverse problems in more intricate double-beam systems.Source: §IV · p. 6

A. Timoshenko Beam Forward Problem

The section evaluates PINNs for a forced Timoshenko beam, which models both transverse displacement and cross-sectional rotation. The reported predictions have low errors and outperform three comparison methods.

A. Timoshenko Beam Forward Problem

  • Timoshenko theory represents beam dynamics with coupled PDEs for transverse displacement and cross-sectional rotation, including transverse shear effects.Source: A. Timoshenko Beam Forward Problem · p. 6
  • Displacement error is 3.3e-4 percent and rotation error is 2.8e-3 percent; maximum deflection is 1.44 and maximum rotation is 0.32.Source: A. Timoshenko Beam Forward Problem · pp. 6–7
  • PGNN had approximately 3.486727% error, and its displacement and rotation predictions both had higher errors than PINN; gPINN also showed higher relative errors than PINN.For gPINN, high-order derivatives in the loss function made autodifferentiation and backpropagation challenging. PGNN’s acceleration data were available only at discrete interior locations, and rotation prediction may have suffered from error propagation.Source: A. Timoshenko Beam Forward Problem · p. 7
  • PINNs predict Timoshenko displacement and rotation more accurately than FDM, PGNN, and gPINN, using 16,000 training points versus FDM’s 30,000.Source: A. Timoshenko Beam Forward Problem · p. 7

B. Timoshenko Beam Inverse Problem

This section estimates a Timoshenko beam’s material parameter from displacement and rotation observations while treating the applied force as known. It reports an estimate near the true parameter and notes practical concerns for inverse training.

B. Timoshenko Beam Inverse Problem

  • The applied force is known and α is the only unknown; the observation data carry 10−3 percent displacement error and 10−4 percent rotation error.The additional data comprise 5,000 points at five beam positions.Source: B. Timoshenko Beam Inverse Problem · p. 8
  • With α = 1 as the reference, PINN estimates α = 1.0136 at t = 0.5 from displacement and rotation data.Source: B. Timoshenko Beam Inverse Problem · p. 8
  • The DNN estimate is α = 0.6124, compared with PINN’s α = 1.0136; the authors report PINN as more accurate for this inverse problem.Source: B. Timoshenko Beam Inverse Problem · p. 8
  • Noisy data may prevent convergence, so filtering or preprocessing may be needed; repeated runs can also yield different solutions because optimizers reach different local minima.The authors recommend empirically increasing training points to find a satisfactory minimum for the problem.Source: B. Timoshenko Beam Inverse Problem · p. 8

C. Euler-Bernoulli Double-Beam Forward Problem

The section applies PINNs to two parallel Euler-Bernoulli beams coupled by a Winkler foundation. It reports low displacement and derived-quantity errors for the forward solution.

C. Euler-Bernoulli Double-Beam Forward Problem

D. Timoshenko Double-Beam Forward Problem

The section models two Timoshenko beams with coupled displacement and rotation fields. It examines forward predictions under different training-point counts and reports small absolute errors across both beams.

D. Timoshenko Double-Beam Forward Problem

E. Timoshenko Double-Beam Inverse Problem

The Timoshenko double-beam inverse experiments estimate an unknown model parameter and applied force from displacement and rotation data. The force is recovered across space and time even with noisy observations.

Timoshenko Double-Beam Inverse Problem

V. CONCLUSIONS

The conclusions report accurate forward and inverse solutions for complex beam systems, including under noisy inverse-problem data. The authors also identify computational cost and generalizability as areas for future work.

V. CONCLUSIONS

  • The framework uses nondimensionalization to aid PINN learning for complex beam systems and solves forward and inverse problems for single and double beams.Source: §V · p. 13
  • Displacement error did not increase with model complexity; it decreased by an order of magnitude for double-beam versus single-beam systems under both beam theories.Bending-rotation error was comparable for single and double Timoshenko beam systems.Source: §V · p. 13
  • For the double Timoshenko beam, the unknown function was approximated over the whole space-time domain with less than 0.05% error, while the single-beam parameter error was less than 3%.Source: §V · p. 13
  • For the Euler-Bernoulli double-beam system, derived velocity, acceleration, and bending moment were approximated with less than 2e-2% error despite not being directly trained.Source: §V · p. 13
  • With 20% Gaussian noise, the force function was discovered with less than 0.2% error; with 1600 training points, double Timoshenko beam displacement had less than 5e-3% error.Both results cover the entire space-time domain where specified.Source: §V · p. 13
  • Future work could address stochasticities, reduce computational cost, improve generalizability, and extend estimation to varied input forces and vibration modes.Source: §V · p. 13

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