P HYSICS -I NFORMED N EURAL N ETWORKS WITH A PPLICATION IN
C OMPUTATIONAL S TRUCTURAL M ECHANICS

Aryan VermaThe Ohio State University
Columbus, OH 43210
verma.306@buckeyemail.osu.edu

Dineshkumar Harursampath
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru, India
dineshkumar@iisc.ac.in

Rajnish Mallick*
Department of Mechanical Engineering
Thapar Institute of University
Patiala, India
rajnish.mallick@thapar.edu

Prasant Sahay
Indian Institute of Science
Bengaluru, India
prasantsahay@iisc.ac.in

Krishna Kant Mishra
Indian Institute of Science
Bengaluru, India
kmkrishna@iisc.ac.in

A BSTRACTStructural mechanics is ubiquitous in the mechanical, aerospace, civil, and biological sciences. With
the advent of machine intelligence technology, it is imperative to study computational structural
mechanics from a machine intelligence standpoint. Helicopter rotor blades are simulated as cantilever
beams, and their complete mechanical behaviour understanding, such as deflections under various
loading conditions, is critical for rotorcraft mechanics. This investigation studies the deflection ofbeams using state-of-the-art physics-informed neural networks (PINNs). An Euler-Bernoulli beamtheory-based beam deflections analytical solution is compared with the PINNs solution. A good
convergence is observed between PINNs and closed-form results. We would like to extend thiswork to Timoshenko beam theory and other dynamical systems in computational structural dynamics(CSD) to develop a digital twin for aerospace and mechanical sciences systems.
Keywords Physics Informed Neural Networks · Computational Structural Mechanics · Neural Networks · SolidMechanics · DeepXDE

International Conference on Scientific Computing and Machine Learning 2024 (SCML2024)

Physics-Informed Neural Networks with Application in Structural Mechanicslearning architecture.1 . It can be thought of two parameters, where the weight is simply the variable parameter,
Physics-informed neural networks (PINNs) have emerged which is what is associated with the input, while the biases
as a groundbreaking tool blending deep learning and phys- are constant values which is what the network adds to theical modeling in the ever-evolving engineering and com- Input before passing it onto the Activation Function.
putational science landscape. PINNs represent a novelX
paradigm in which neural networks are trained to solve
W eight ∗ Inputs + Bias
supervised learning tasks while respecting the underlying physical laws described by general nonlinear partial The key differentiation between the PINNs methodology
differential equations1, 2 . This fusion of physics-based and conventional neural network paradigms principally
constraints with the versatility of neural networks opens manifests within the framework of the loss function. In
new avenues for solving complex engineering problems, PINNs, the composition of the loss function is intricately
particularly in aerospace engineering.
expanded to include not only the Mean Squared Error
This paper primarily focuses on applying PINNs in beam (MSE) loss, typical of standard neural networks but alsomechanics, a fundamental area in structural engineer- incorporates the physics loss, i.e. boundary conditions
ing with profound implications in aerospace design and loss and the derivative loss. This enriched loss function
analysis1-4 . Traditionally approached through analytical undergoes an iterative optimization, a process elegantlyand numerical methods, beam mechanics stands to gain illustrated in Figure 1 below.
significantly from integrating PINNs, offering enhanced
accuracy and efficiency in simulations5-6 . The versatilityof PINNs, particularly in embedding physical laws into
the learning algorithm, ensures that the solutions are notjust data-driven but also explainable. This aspect is crucial in aerospace engineering, where adherence to physicalrealities is non-negotiable.

1

Introduction

Rotorcraft researchers are continuously exploring noveland computationally efficient algorithms to solve criticalaeromechanics problems, such as helicopter vibration reduction using smart trailing edge flaps at the rotor blades7 .
Rotor blades can be simulated as cantilever beams for modeling and simulations. There is a constant quest worldwide
to develop and integrate novel computational methods toadvance rotor mechanics further and computational structural mechanics.

Figure 1: The structure of a Physics Informed Neural
Network
The network ingests spatial inputs x, applies a feedforward
neural network (FNN) with three hidden layers and 20 neurons each, using the hyperbolic tangent activation function
and Glorot normal initialization. Along with the meansquared error loss, the PINNs Network also takes into account the Boundary Conditions, the Governing Equation,
for which the Network uses the 1st, 2nd, and 3rd derivates
as well.

The primary objective of this investigation is to explore
the potential of PINNs in various aspects of aerospaceengineering. From structural analysis and aeroelasticity tothermal management and propulsion system optimization,
PINNs offer a versatile tool for tackling complex problems
that are otherwise challenging to solve with traditional
methods. In aerospace engineering, where the systems 3 Physics Explanation
are often subjected to extreme conditions and complex
interactions, PINNs can provide more accurate, efficient, In this study, a cantilever beam, subjected to a uniform
and robust solutions.
distributed load (an analogous special load case of a fixed
wing or rotary wing), illustrated in Figure 2, was considered for the PINNs case-study 8 . Over here, we take L as2

Introduction to the Idea of Physics
Informed Neural Networks (PINNs)

From a mathematical perspective, PINNs2 can be comprehensively appreciated by recognizing that the neuralnetwork’s output is systematically amalgamated with theresidual of the system’s differential equation and corresponding boundary conditions, resulting in a precise solution. This formulation extends the foundational prin- Figure 2: Cantilever Beam with Uniformly Distributed
ciple of neural networks, wherein the input features are Load 8
linearly transformed through weights and biases, thereby
integrating the physical laws governing the system into the the Length of the Beam. To get the governing equation,2

Physics-Informed Neural Networks with Application in Structural Mechanics

the bending moment in the beam at distance x (the point’s Before reaching the final result, a comprehensive parametlocation along the beam) from the fixed support can be ric study is conducted to identify the best number of epochs
calculated as,for convergence and different activation functions are also
explored, such as ReLU, tanh, ELU, Sigmoid, and Softmax. Various optimizers are also explored, such as Adam,
22−qLqx
M=+ qLx −(1) RMSProp, and Stochastic Gradient Descent, and different
22learning rates are also investigated as part of the parametric
Where q is the uniform load intensity. The differential study. The result finally matched the exact Solution of theequation v” for beam deflection according to the Euler- Cantilever beam equation at 2000 epochs, using the Adam
optimizer and the tanh activation function with a learning
Bernoulli theory is :rate of 0.01 with three hidden layers in the Code.
M (x)′′v =−(2) It can be concluded that PINNs are capable of fast andEI
accurate computations for computational mechanics probM(x) is the bending moment along the beam, E is our lems and would be relevant to aerospace structures, other
Elastic Modulus, which indicates how stiff the beam is, mechanical sciences geometric primitives, and industrialand I is the 2nd Moment of Inertia.The first equation can systems. In the future, these systems are capable of being
be plugged into the beam deflection curve differential at the forefront of the industry, and we can prove throughthis research that any PINNs network will reproduce anequation8 , and the combined equation is:
accurate solution for any system if calibrated properly.
−qL2
qx2′′
EIv =+ qLx −(3)22Integrating the above equation and using the boundary
conditions, the angle of rotation v’ is calculated as follows:v′ =−qx(3L2 − 3Lx + x2 )
6EI

(4)

The following boundary conditions are considered:v(0) = 0,v ′ (0) = 0−qL3
6EI−qL4v(L) =8EIv ′ (L) =(5)(6)(7)

Upon integration for the deflection equation with theboundary conditions, which is the equation compared tothe real solution, the equation for the deflection of thisbeam is:−qx2v=(6L2 − 4Lx + x2 )(8)
24EI4Figure 3: The Graph of the PINNs Solution vs the ExactSolutionResults and Discussion

In this study, a beam measuring 2.7 meters long was considered and is subjected to a uniform load of 60 kilo-Newtons,
Young’s Modulus is equal to:
200 × 109 Pa(9)

and Moment of Inertia is equal to:
0.000038929334

kg · m2(10)With the geometric parameters and PINNs formulation inplace, the exact solution and PINNs solution is illustrated
in Figure 3, where it can be observed that the PINNs network reproduces exactly the same solution as the exact
solution of the Equation.

Figure 4: Convergence of the PINN model over 20,000
training iterations, showing the decrease in training andtest losses with the L2 relative error as a test metric.
3Physics-Informed Neural Networks with Application in Structural Mechanics

6In Figure 4, we present the convergence trends of thePhysics-Informed Neural Network (PINN) over 20,000
training steps (epochs). The blue line represents the training loss, and the orange line represents the test loss. Bothlosses demonstrate a steep decline in the initial training
phase, followed by a plateau, indicating the model’s learning and subsequent stabilization. The green line represents
the test metric, which predicts how different the real values
are from the predicted values of the network, specificallythe L2 relative error, which is a normalized measure of themodel’s predictive accuracy across the test dataset6 .

This work is financially assisted by the Science and Engineering Research Board (SERB), DST, India, under
MATRICS Scheme, File number: MTR/2022/001029.

7

L2 relative error =N
2i=1 (ypred,i − ytrue,i )qPN
2i=1 ytrue,iReferences

1. Lu, L., Meng, X., Mao, Z., and Karniadakis, G. E.
(2021). DeepXDE: A deep learning library for solving
differential equations. SIAM review, 63(1), 208-228.

The L2 relative error is mathematically defined as:qP

Acknowledgement

2. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019).
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving(11) nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707."
3. Katsikis, D., Muradova, A. D., and Stavroulakis, G.
E. (2022). A Gentle Introduction to Physics-Informed
Neural Networks, with Applications in Static Rod andBeam Problems. Journal of Advances in Applied andComputational Mathematics, 9, 103-128.

where N is the number of samples in the test dataset, ypred,iis the model’s prediction for the i-th sample, and ytrue,iis the corresponding true value. The L2 relative error
quantifies the overall deviation of the model’s predictions
from the true values, normalized by the magnitude of thetrue values to facilitate comparison across different scales.

4. Markidis, S. (2021). The old and the new: Can physicsinformed deep-learning replace traditional linear solvers?.
In the context of Figure 4, the declining trend of the L2 rel- Frontiers in big Data, 4, 669097.
ative error alongside the losses reflects the model’s increas- 5. Bazmara, M., Silani, M., and Mianroodi, M. (2023,ing accuracy. Notably, the green line exhibits fluctuations, March). Physics-informed neural networks for nonlinear
particularly in the latter half of the training process. These bending of 3D functionally graded beam. In Structuresfluctuations in the L2 relative error could be indicative of (Vol. 49, pp. 152-162). Elsevier.
variations in the model’s performance on the test dataset
at different iterations and warrant further investigation to 6. Kag, V., and Gopinath, V. (2023). Physics-informed neural network for modeling dynamic linear elasticity. arXiv
ensure the robustness and reliability of the predictions.
preprint arXiv:2312.15175.

5

7. Mallick, R., Ganguli, R., and Bhat, M. S. (2015). Robust
design of multiple trailing edge flaps for helicopter vibration reduction: a multi-objective bat algorithm approach.
Engineering Optimization, 47(9), 1243-1263.

Conclusion

In this investigation, the state-of-the-art physics-informed
neural networks (PINNs) technique is studied for appli- 8. Gere, J. M., and Goodno, B. J. (2012). Mechanics ofcations in computational mechanics, in particular beam materials. Cengage learning, Pg. 491-492.
mechanics. A good convergence is observed between
PINNs, and closed-form results of the Euler-Bernoulli
theory-based beam deflection analytical solution. PINNs
solutions are also computationally efficient; the results
took an average of 19.5 seconds to come up with a solution
to the equation, proving that the Physics Informed NeuralNetworks (PINNs) give effective and reasonably fast results for mechanical and aerospace systems with optimal
hyperparameters. We used 100 residual points for testing
the residual and fixed 32 residual points for training, which
proved effective, with 3.71e-08 being the best train loss and2.72e-08 being the best test loss. Integration of data-driven
and physics is an ideal candidate to develop a digital twin
for aerospace and mechanical sciences systems. In future,
this work will be extended to Timoshenko beam theory
and other dynamical systems in computational structuraldynamics (CSD) and also develop an optimization routine
for automatic hyperparameter tuning.
4