Source-linked AI summary
Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios
Chen Xu, Ba Trung Cao, Yong Yuan, Günther Meschke
TL;DR
The paper addresses inverse structural analysis with limited displacement data by combining physics-informed neural networks, uncertainty-weighted multi-task learning, and transfer learning. The approach predicts unknown loads and accelerates convergence across elasticity and hyperelasticity problems, while remaining applicable to geometrically scaled structures.
Problem
Inverse structural analysis requires predicting unknown external loads from limited monitoring data while solving elasticity and hyperelasticity problems.
Method
The paper uses boundary-condition-learnable PINNs with non-dimensionalization, homoscedastic uncertainty weighting, and offline-to-online transfer learning.
Results
The approach successfully characterized unknown structural loads from noisy observational data and accelerated convergence, including convergence within 1% error after 6900 epochs in one hyperelastic plate example.
Takeaways & Limitations
Transfer learning based PINNs can support inverse analysis and surrogate modeling for engineering structures with geometric scaling and varied loading scenarios.
Takeaways & Limitations
The study assumes external loads are uniformly distributed over specified regions and does not cover all practical boundary types, such as concentrated loads.
Abstract
from arXiv · showhide
Recently, a class of machine learning methods called physics-informed neural networks (PINNs) has been proposed and gained prevalence in solving various scientific computing problems. This approach enables the solution of partial differential equations (PDEs) via embedding physical laws into the loss function. Many inverse problems can be tackled by simply combining the data from real life scenarios with existing PINN algorithms. In this paper, we present a multi-task learning method using uncertainty weighting to improve the training efficiency and accuracy of PINNs for inverse problems in linear elasticity and hyperelasticity. Furthermore, we demonstrate an application of PINNs to a practical inverse problem in structural analysis: prediction of external loads of diverse engineering structures based on limited displacement monitoring points. To this end, we first determine a simplified loading scenario at the offline stage. By setting unknown boundary conditions as learnable parameters, PINNs can predict the external loads with the support of measured data. When it comes to the online stage in real engineering projects, transfer learning is employed to fine-tune the pre-trained model from offline stage. Our results show that, even with noisy gappy data, satisfactory results can still be obtained from the PINN model due to the dual regularization of physics laws and prior knowledge, which exhibits better robustness compared to traditional analysis methods. Our approach is capable of bridging the gap between various structures with geometric scaling and under different loading scenarios, and the convergence of training is also greatly accelerated through not only the layer freezing but also the multi-task weight inheritance from pre-trained models, thus making it possible to be applied as surrogate models in actual engineering projects.
1. Introduction
PINNs integrate physical laws, constraints, observational data, and mathematical models to address scientific-computing problems, including inverse analysis. This paper targets underexplored inverse applications by combining uncertainty-weighted multi-task learning with transfer learning for structural engineering.
- Physics-informed neural networks: PINNs solve PDEs by embedding physical descriptions such as laws and constraints into neural networks, without requiring big data.They can integrate observational data with mathematical models for forward and inverse problems.
- Research gap: Existing PINN research has focused mainly on forward problems, while inverse-analysis performance remains insufficiently investigated.The paper identifies this as a gap motivating its structural-engineering application.
- Research gap: Many existing approaches add gradient calculations or operations during training, increasing computational complexity.This limitation is stated alongside the limited investigation of inverse analyses.
- Proposed approach: The paper introduces homoscedastic-uncertainty multi-task learning to weight learning objectives for linear elasticity and incompressible Neo-Hookean hyperelasticity.The method is intended to enhance accuracy and accelerate convergence at low computational cost.
- Proposed approach: Unknown boundary conditions are treated as learnable parameters so PINNs can predict structural loads from monitoring data.The approach is designed for inverse analysis when sufficient displacement or other monitoring data are available.
- Proposed approach: Transfer learning fine-tunes pre-trained PINNs for engineering structures of different sizes and loading scenarios, using inherited network and task parameters.The paper motivates transfer learning by the geometric similarity of many engineering structures and its ability to accelerate training for new tasks.
2. Methodology
The methodology embeds governing equations and boundary conditions into PINN losses, then extends the framework with uncertainty-weighted multi-task learning and transfer learning for elasticity inverse analyses.
- 2.1. PINNs: Physics-informed neural networks: PINNs approximate displacement fields with feed-forward networks whose trainable weights and biases are optimized against physics and boundary-condition losses.The PDE loss evaluates differential-equation residuals at domain points, while the boundary loss evaluates boundary-operator residuals.
- 2.3. Hyperelasticity with non-dimensionalization: The elasticity formulation uses dimensionless variables and equations for linear elasticity and incompressible Neo-Hookean hyperelasticity.For hyperelasticity, the network additionally outputs scaled hydrostatic pressure and enforces incompressibility through J = det(F) = 1.
- 2.4. Multi-task learning using uncertainty: Uncertainty weighting learns task weights jointly with network parameters so different loss objectives are balanced during optimization.The logarithmic term regularizes the task-weight parameters and prevents them from becoming too large.
- 2.5. Proposed approach: Transfer learning based boundary-condition-learnable PINNs: The proposed model customizes its loss with multiple tasks and learnable boundary-condition parameters for elasticity analysis and load prediction.The total parameter set includes network parameters, task noise parameters, and predicted external loads.
- 2.5. Proposed approach: Transfer learning based boundary-condition-learnable PINNs: Transfer learning freezes the initial three layers and fine-tunes remaining parameters with a smaller learning rate for geometrically similar structures and changed scenarios.Pretrained task weights are inherited to balance loss terms at the beginning of retraining, while fewer parameters require updating.
3. Benchmark examples
Three benchmark problems evaluate non-dimensionalization, uncertainty-weighted multi-task learning, and robustness to noisy displacement data in structural PINNs.
- 3.1. Prediction of internal pressure acting on a thick elastic cylinder: The thick-cylinder benchmark compares plain, scaled, self-adaptive, and uncertainty-weighted PINNs for identifying unknown internal pressure.The cylinder uses an analytical displacement solution as the target, with P = 20, E = 1 × 10^5, and ν = 0.3.
- 3.1. Prediction of internal pressure acting on a thick elastic cylinder: P = 19.929 (error: 0.355%) is obtained with uncertainty weighting, while scaled PINN gives P = 19.569 (error: 2.155%).Both multi-task approaches reduce the maximum L2 error of field outputs to less than 1% compared with scaled PINN.
- 3.1. Prediction of internal pressure acting on a thick elastic cylinder: Uncertainty weighting reaches within 1% error at epoch 13500, versus epoch 24500 for SA-PINN.Its additional parameters depend on the number of tasks rather than the number of training points, reducing computational complexity.
- 3.2. Prediction of line load acting on a square hyperelastic plate: The hyperelastic square-plate benchmark estimates a learnable uniform line load from FEM displacement data under a clamped boundary condition.The 2 × 2 incompressible Neo-Hookean plate uses P = 20, E = 1000, and ν = 0.5.
- 3.2. Prediction of line load acting on a square hyperelastic plate: 6900 epochs are required for uncertainty-weighted PINN to reach within 1% error, compared with 8500 for scaled PINN and 7500 for SA-PINN.Final predictions are P = 14.858 (error: 0.947%), P = 14.847 (error: 1.020%), and P = 14.886 (error: 0.760%), respectively.
- 3.3. Prediction of tensile loads acting on a 3D elastic beam: In the noisy 3D beam benchmark, uncertainty weighting predicts P = 2.0022 (error: 0.11%) from noisy displacement fields.All three methods have similar accuracy, while uncertainty weighting converges fastest within 1% error; its smallest data-loss weight prioritizes physics loss.
- 3.3. Prediction of tensile loads acting on a 3D elastic beam: Physics laws prevent overfitting to noisy data: MSEdata stops decreasing after 5000 epochs, while MSEtrue drops to 10^-7.The result indicates identification of the correct solution despite noise of around 10%.
4. Model transfer for inverse analyses across computational domains
The study transfers PINN knowledge from simplified offline problems to geometrically scaled plates and tunnel load-inference problems using limited displacement data. Transfer learning preserves prediction accuracy while accelerating convergence, and tunnel-load predictions remain accurate across similar and unlike loading scenarios.
- Geometric scaling: Transfer learning adapts a pre-trained square hyperelastic-plate PINN to rectangular plates with 4×2 and 2×4 geometries.The rectangular cases are treated as scaled versions of the square benchmark.
- Geometric scaling: Only 20 randomly selected displacement data points are used for each rectangular-plate inverse analysis.The online models freeze the first three layers and fine-tune the remaining parameters at a learning rate of 0.0005.
- Geometric scaling: Transfer learning accelerates convergence without losing accuracy for the rectangular-plate line-load predictions.Freezing two or three layers produces good predictions, whereas freezing four layers fails to converge within the expected epochs.
- Geometric scaling: The number of frozen layers trades computational efficiency against network expressivity and accuracy.The authors report that more frozen layers reduce expressivity, while domain adaptation is not considered.
- Tunnel loading: The tunnel inverse problem estimates external loads from displacement measurements at limited monitoring locations under simplified vertical and lateral soil pressures.The 2D tunnel-ring case uses P1 = 100kN and P2 = 80kN as directly applied pressures.
- Tunnel loading: The tunnel PINN predicts P1 = 100.479kN with 0.479% error and P2 = 80.968kN with 1.210% error.The reported displacement-field L2 errors are 11.250% in X and 10.060% in Y for the 2p case.
- Tunnel loading: Average prediction errors for all four acting loads in tunnel cases 4p a and 4p b are less than 3%.Comparable accuracy is obtained for both similar and unlike loading scenarios relative to the pre-trained case.
- Tunnel loading: Transfer learning remains robust to Gaussian noise and reduces training cost through parameter freezing and retained physical characteristics.The reported computing time without task-weight transfer is around 17 minutes.
5. Conclusions
The paper concludes that non-dimensionalization, uncertainty-weighted multi-task learning, and two-stage transfer learning improve PINN-based inverse analysis. The approach predicts unknown structural loads from noisy observations across scaled geometries and more complicated loading scenarios, while reducing data and computational requirements.
- Conclusions: Non-dimensionalization and homoscedastic uncertainty weighting significantly improve PINN performance.The weighting allocates relative importance across different learning tasks.
- Conclusions: PINNs can characterize unknown structural loads from sufficiently numerous noisy observations when physics laws regularize the inverse problem.Unknown boundary conditions are represented as learnable parameters.
- Conclusions: The two-stage transfer-learning approach predicts loads for scaled geometries and loading scenarios more complicated than the pre-trained boundary conditions.The offline model supplies physics features that guide accurate online predictions.
- Conclusions: Transfer learning reduces both the required observation-data volume and computational cost.The authors present this reduction as part of the approach’s applicability to engineering inverse analysis.
- Conclusions: The multi-task method adds only a small number of learnable parameters and provides fast convergence, although it is not the most accurate method.Its task weights have a mathematical interpretation as homoscedastic uncertainties.
- Limitations: The study assumes external loads are uniformly distributed over specific regions and does not cover all engineering boundary types.Localized partial-surface loads may create sharp stress gradients that degrade PINN performance.