Physics Meets Machine Learning to Predict the Full Seismic Response of

Engineers have long struggled to predict how buildings will behave during an earthquake. Detailed computer simulations of steel frames and combined steel-and-concrete walls follow the laws of physics, but they take a long time to run and depend on many assumptions. Machine learning models work much faster, but they usually predict only one result, such as the greatest load a structural part can carry. They also often work like black boxes, making their decisions hard to understand. A new study tackles both problems with a machine learning method that uses the laws of physics. It predicts the full response of steel frames and composite walls that absorb earthquake energy, including how they lose stiffness and strength and how they absorb energy. It can do this even when trained on very little experimental data. When tested against laboratory experiments involving repeated loading, the method achieved scores above 0.90 for every specimen tested. It performed better than both standard machine learning models and a method that combines machine learning with numerical simulations.

The study looks at prefabricated light steel frames and systems that combine steel frames with composite walls. The researchers build well-known mechanical rules directly into the way their neural networks learn. Instead of letting a network freely fit measurements of load and movement during repeated loading, they add physics-based rules to the loss function, which measures how far the network's predictions are from its targets. These rules cover the Bouc–Wen model of how materials respond to repeated loading, stable behavior as loading changes direction, limits on energy loss, elastic recovery, and damage to concrete as it bends or stretches and gets compressed. This means predictions must follow the physical rules that govern structures under repeated loading at relatively low levels of cycling. The authors describe their approach as a special kind of machine learning that combines data-based learning with physics-based rules in its loss function. Their goal is to keep predictions accurate and understandable even when only a small amount of training data is available. This is important in structural testing because building and testing full-size specimens is expensive and takes a long time.

The method begins with the standard equation of motion for a structure under an external load. In this equation, the forces caused by mass and acceleration, viscous damping, and the structure's restoring force balance the applied load. The restoring force has two parts: an elastic part and a hysteretic part. The hysteretic part is controlled by an internal variable called hysteretic displacement, which changes according to a set of differential equations in the Bouc–Wen model. Because these equations cannot directly provide a solution for this internal variable, the researchers integrate its evolution equation and use a long short-term memory network, or LSTM, to represent it. The network learns the shape parameters of the hysteresis model as part of the physics-based loss function. The researchers also rewrite the restoring-force equation as a differential equation. Its changes are linked to displacement, energy loss, and the hysteretic variable through a function that the network learns.

The researchers chose an LSTM for a specific reason. Standard recurrent neural networks can struggle to learn relationships that last a long time in a sequence because their training gradients can become too small or too large. LSTMs address this problem with three gates: a forget gate, an input gate, and an output gate. These gates control which information is kept or discarded in the network's memory as it processes each step in a sequence. This makes LSTMs a good fit for hysteresis, where the force a specimen resists at a particular displacement depends on its entire previous loading history. The LSTM receives displacement histories, loading sequences, and structural properties as inputs. It has two hidden LSTM layers containing 64 and 32 units, followed by a fully connected output layer. It uses the tanh activation function and the Adam optimizer, with a learning rate of 0.001, a batch size of 32, and 2000 training epochs.

A second network, called a multilayer perceptron (MLP), predicts the restoring force. This is a feedforward network in which each layer is fully connected to the next. Using Leibniz's rule for differentiating an integral, the researchers derive a differential equation for the restoring force that links it to displacement, energy loss, and the hysteretic variable. They then use an MLP to learn the function inside that equation. The network has three hidden layers with 64, 32, and 16 neurons and uses the ReLU activation function. Like the LSTM, it is trained with Adam for 2000 epochs. The data are scaled using min–max normalization and divided into 80 percent for training and 20 percent for validation. The researchers combine the usual data-based loss with seven separate physics-based loss terms. These check whether the hysteretic displacement follows the Bouc–Wen model, whether the restoring force behaves stably when displacement reverses direction, whether energy loss is non-negative, whether stored energy steadily decreases after the elastic stage, whether the structure returns to zero displacement under the yield load, and whether plastic behavior follows Drucker's hypothesis. The seventh term checks concrete damage in tension and compression. These damage measures are calculated from material behavior models calibrated with experiments and from numerical simulation results, rather than being predicted directly by the network.

The dataset came from repeated-loading tests on two groups of specimens. The first group contained six bolted beam–column joint specimens: one standard bolted connection and five L-shaped joints. Their columns included square steel tubes, square steel tubes filled with concrete, and H-shaped steel columns. These were paired with H-shaped steel beams. The second group contained six half-scale light steel frame specimens with energy-dissipating composite shear walls. Each specimen combined a steel tube frame with columns filled with recycled concrete, H-shaped steel beams, and a recycled concrete wall panel containing a hidden steel support. The support type varied between reinforced steel and steel plate. Bar spacing ranged from 100 to 200 millimeters, while the reinforcement ratio ranged from 0.33 percent down to 0.16 percent. Each loading history was divided into evenly spaced time steps, with about 360 to 450 steps per history. Together, the data produced about 5000 sequential data points. The researchers divided the data by specimen, assigning 70 percent to training, 15 percent to validation, and 15 percent to testing.

In tests against these specimens, the physics-informed method performed better than both a standard machine learning model and a machine learning-based numerical simulation method that corrects for contact stiffness. For the six frame specimens, the new method achieved R² scores between 0.954 and 0.970, with a reported minimum of 0.95 and maximum of 0.97. The machine learning-based simulation method reached a maximum of 0.939, while standard machine learning scores were about 0.902 to 0.918. On average, the new method raised the R² score from 0.929 for the simulation-based method to 0.964. It also reduced root mean square error by about 45.8 percent and mean absolute error by about 47.1 percent. For the six joint specimens, the new method kept the error in total energy dissipation below 4 percent. By comparison, standard machine learning had errors of about 12.1 to 14.1 percent, while the machine learning-based simulation method had errors of about 9.0 to 10.8 percent.

The composite wall specimens were more complicated, so the predictions were slightly less accurate, although they still performed well. Across the six walls, the new method achieved R² scores from 0.902 to 0.934. The machine learning-based simulation method reached a maximum of 0.872, while standard machine learning scored about 0.818 to 0.841. The new method also reduced the error in total energy dissipation to about 4.6 to 5.9 percent. Standard machine learning errors were about 16.9 to 19.0 percent, and simulation-based method errors were about 12.5 to 14.1 percent. The predicted curves showing force against displacement closely followed the curves measured in the experiments. They captured energy absorption and changes in the structure's ability to carry loads as it moved between elastic and plastic behavior. The standard data-driven network visibly failed to reproduce these patterns. The results also reflected the two separate lines of seismic protection observed in the frame-and-wall system during the experiments.

The authors say the method also reduces much of the work needed for traditional finite element modeling. It does not require dividing the structure into a mesh, setting contact rules between the steel frame and the walls, explicitly linking the degrees of freedom at welded connections, or assuming simplified elastic-plastic material behavior. Instead, the prediction model is built directly from measured geometry, loading conditions, and material information. Because physical laws are included as rules that the network must follow, its predictions remain understandable and consistent with physics even when the training dataset is small. This helps reduce overfitting and the lack of transparency that can affect models trained only on data. The researchers present the method as a tool for assessing how steel frames and energy-dissipating composite wall systems may perform during earthquakes. They also see it as a step toward bringing structural engineering and artificial intelligence closer together. However, the slight drop in accuracy for the more complicated wall systems shows that complex physical behavior remains a challenge even for machine learning methods guided by physics.

Subject of Research: Machine learning guided by physical laws to predict the full earthquake response of steel frames and composite wall structures under repeated loading

Article Title: A hysteretic performance prediction model for steel–concrete composite structures integrating physical processes and machine learning

Article References: Wei, D., & Suizi, J. (2026). A hysteretic performance prediction model for steel–concrete composite structures integrating physical processes and machine learning. Results in Engineering, 32, Article 113571. https://doi.org/10.1016/j.rineng.2026.113571

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113571

Keywords: machine learning guided by physics, earthquake performance, hysteresis, Bouc–Wen model, LSTM, steel–concrete composite structures, energy dissipation, prefabricated structures, structural engineering, neural networks, learning from small datasets, loss of stiffness