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Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures
Ruiyang Zhang, Yang Liu, Hao Sun
TL;DR
Nonlinear structural metamodeling is difficult when simulations are expensive and data are scarce, incomplete, or noisy, particularly because latent hysteretic variables may be unmeasured. The paper addresses this gap with physics-informed multi-LSTM networks that encode governing equations, state dependencies, and hysteretic relationships into the architecture and loss function. Across two numerical examples, the approach captures structural dynamics and improves robustness and prediction accuracy over classical LSTM and non-physics-guided data-driven models, while its demonstrated context remains structural seismic response prediction.
Problem
Nonlinear structural metamodeling requires handling expensive simulations, scarce or incomplete data, and latent hysteretic variables that are not directly measured.
Method
The paper connects multiple deep LSTM networks with a graph-based differentiator and embeds governing equations, state dependencies, and hysteretic relationships into physics-informed training losses.
Results
The proposed PhyLSTM2 and PhyLSTM3 networks accurately capture structural dynamics with scarce data and produce more robust, accurate predictions than classical LSTM without embedded physics.
Takeaways & Limitations
Embedding physics enables nonlinear structural metamodeling within a feasible solution space while reducing reliance on large training datasets and improving prediction robustness.
Takeaways & Limitations
The demonstrated metamodels target structural seismic response prediction and require adaptation of the physics component for other structural systems.
Abstract
from arXiv · showhide
This paper introduces an innovative physics-informed deep learning framework for metamodeling of nonlinear structural systems with scarce data. The basic concept is to incorporate physics knowledge (e.g., laws of physics, scientific principles) into deep long short-term memory (LSTM) networks, which boosts the learning within a feasible solution space. The physics constraints are embedded in the loss function to enforce the model training which can accurately capture latent system nonlinearity even with very limited available training datasets. Specifically for dynamic structures, physical laws of equation of motion, state dependency and hysteretic constitutive relationship are considered to construct the physics loss. In particular, two physics-informed multi-LSTM network architectures are proposed for structural metamodeling. The satisfactory performance of the proposed framework is successfully demonstrated through two illustrative examples (e.g., nonlinear structures subjected to ground motion excitation). It turns out that the embedded physics can alleviate overfitting issues, reduce the need of big training datasets, and improve the robustness of the trained model for more reliable prediction. As a result, the physics-informed deep learning paradigm outperforms classical non-physics-guided data-driven neural networks.
1. Introduction
Structural simulations support analysis and design but become costly for nonlinear systems under dynamic loads, especially when many runs are needed. Metamodels and neural networks offer faster prediction, yet complex time-series behavior and scarce or incomplete data remain challenges.
- Finite element and other numerical simulations are widely used for structural analysis and design under dynamic loading.
- Computational costs rise substantially for nonlinear hysteretic structures when optimization or uncertainty analyses require numerous simulations.
- Metamodels replace time-consuming simulations with faster input-output approximations, but traditional regression and response-surface methods may lack accuracy for complex systems.
- Artificial neural networks can improve prediction accuracy and capture nonlinear input-output relationships, while deep recurrent models are promising for dynamical-system time histories.
- Black-box models depend strongly on representative labeled data and face accuracy, generalizability, interpretability, and training challenges when data are scarce, incomplete, or noisy.
- The paper proposes physics-informed double- and triple-LSTM architectures and evaluates them on nonlinear systems with rate-independent and rate-dependent hysteresis.
2. Physics-informed Multi-LSTM Network for Metamodeling
The paper embeds structural physics into interconnected deep LSTM networks to model nonlinear dynamic responses, including the latent hysteretic parameter that ordinary data-driven LSTMs cannot observe.
- Physics formulation: The reduced-fidelity equation of motion maps ground acceleration to displacement, velocity, acceleration, and nonlinear restoring force in structural dynamics.The restoring force includes an auxiliary hysteretic parameter and is represented through a mass-normalized latent function of state variables.
- Physics formulation: The state space comprises displacement, velocity, and hysteretic parameter, but the latter is typically non-observable and cannot be directly learned from standard measurements.This makes direct LSTM metamodeling of the full state space inapplicable without additional physics information.
- Network design: PhyLSTM2 uses two deep LSTMs to model state variables and nonlinear restoring force, whereas PhyLSTM3 adds a third LSTM for the hysteretic parameter equation.PhyLSTM3 also uses a system-variable library Φ, which may be simplified when prior knowledge is unavailable.
- Network design: A physics-informed multi-LSTM architecture connects a data-modeling LSTM with one or two physics-modeling LSTMs through a graph-based tensor differentiator.The differentiator computes derivatives of state-space outputs so governing constraints can enter training.
- Physics constraints: PhyLSTM3 physics loss enforces state-variable equality, the equation of motion, and the hysteretic parameter equation during optimization.The loss includes a user-defined weighting coefficient η, with η = 1 given as a simple example.
3. Numerical Validation: 3-story Moment Resisting Frame
A synthetic 3-story steel moment-resisting-frame example evaluates PhyLSTM2 and PhyLSTM3 against classical LSTM using earthquake-driven nonlinear response data. Across unseen excitations, the physics-informed models provide accurate and robust displacement, velocity, and latent-state predictions from limited training data.
- Numerical model: The validation uses a full-scale 3-story office building modeled with a single-bay moment-resisting frame and associated structural systems.The prototype is 45.7 m by 45.7 m in plan and 11.43 m in elevation.
- Data generation: A synthetic database is generated from nonlinear time-history simulations under 97 earthquake records selected for the Pomona, California site.Training and validation records provide ground motion, displacement, and velocity data, while prediction data are reserved for testing.
- Data generation: The datasets contain 46 samples with 10,001 time points and are split 0.8/0.2 for training and validation, with additional earthquake samples used during physics-informed training.The arrays contain three output channels compatible with the LSTM networks.
- Prediction performance: Across 760 testing datasets, most PhyLSTM2 and PhyLSTM3 displacement correlation coefficients γ exceed 0.9, whereas classical LSTM performs less robustly.The worst reported correlations are γ = 0.74 for PhyLSTM2, γ = 0.76 for PhyLSTM3, and γ = 0.25 for LSTM.
- Prediction performance: For representative unseen earthquakes, PhyLSTM predictions reproduce displacement magnitudes, phases, and residual drifts, including a PhyLSTM2 case with γ = 0.95.The corresponding reported correlations are PhyLSTM3 γ = 0.95 and 0.89, and LSTM γ = 0.66 and 0.85.
- Physics-informed prediction: Physics-informed networks also predict latent hysteretic states without state measurements and learn hidden patterns obeying governing laws from very limited data.The latent outputs include the hysteretic parameter r or nonlinear restoring force g.
4. Numerical Validation: Bouc-Wen Hysteresis Model
The Bouc-Wen validation compares PhyLSTM2 and PhyLSTM3 on unseen excitations, showing stronger displacement prediction and latent restoring-force estimation for PhyLSTM3.
- Model and setup: The SDOF Bouc-Wen example models rate-dependent hysteresis and compares PhyLSTM2 with PhyLSTM3.The model uses nonlinear parameters α_i, β_i, and n_i in its rate-dependent hysteresis formulation.
- Displacement prediction: PhyLSTM3 produces a larger probability of correlation coefficients near one than PhyLSTM2 for unseen BLWN displacement histories.In the worst scenario, PhyLSTM3 reaches γ = 0.77 compared with PhyLSTM2 at γ = 0.19.
- Displacement prediction: For two example displacement histories, PhyLSTM2 obtains γ = 0.85 and γ = 0.19, while PhyLSTM3 obtains γ = 0.99 and γ = 0.77.These examples compare the two architectures under unseen excitations.
- Latent nonlinear response: PhyLSTM3 predicts the mass-normalized restoring force with γ ≈1 without restoring-force measurements during training.Its predicted hysteresis curves also estimate nonlinear restoring force versus displacement, and over 95% of correlations exceed 0.9 across 97 additional ground motions.
5. Conclusions
The paper proposes physics-informed multi-LSTM metamodels for nonlinear structural seismic responses using limited data. It reports accurate latent-state modeling and performance advantages over classical non-physics-guided LSTM models, while noting that adaptations are needed for other structural systems.
- Contributions: PhyLSTM2 and PhyLSTM3 embed physics constraints in network architectures and the loss function to learn sequence-to-sequence structural behavior from limited data.The framework targets a feasible solution space for nonlinear structural dynamics.
- Contributions: The models can represent non-observable latent nonlinear states, including hysteretic parameters and nonlinear restoring force, when measurements are unavailable.This capability is demonstrated in the paper’s numerical examples.
- Validation: Numerical examples comprise a 3-story moment-resisting frame and an SDOF Bouc-Wen model for nonlinear structural response prediction.The examples cover rate-independent and rate-dependent hysteresis settings.
- Conclusions: The proposed physics-informed multi-LSTM models outperform classical non-physics-guided data-driven neural networks in the reported numerical results.
- Scope: The approaches are presented for structural seismic response prediction and require adaptation of the physics component for other structural-system types.The network architectures should be modified by changing the physics part as needed.