In the identification stage, the researchers reformulated drag coefficient tracking as a Markov decision process and employed the TD3 reinforcement learning algorithm to interact successively with a ballistic simulation environment. This interaction allows the method to capture the time-varying drag coefficient as the projectile decelerates, rather than assuming a fixed parameter value throughout flight.

A key innovation is a cross-trajectory parameter screening and retraining strategy introduced to resolve parameter inconsistency across multiple trajectories. According to the authors, this yields a more consistent drag curve with improved generalization across trajectories, a persistent challenge when a single drag model must serve many different flight paths.

Based on the identified results, the team constructed a simulation-generated trajectory dataset to support the subsequent prediction stage. This close coupling between identification and prediction, the paper notes, ensures that model-based physical consistency benefits the data-driven forecasting process.

For prediction, the researchers built a bidirectional LSTM network with a multi-head attention mechanism, abbreviated BiLSTM–AM, designed to efficiently capture temporal dependencies and nonlinear ballistic features.

The framework was validated against measured trajectory data. The identification module reduced the overall average trajectory reconstruction RMSE by 22.45% compared with the conventional averaging strategy, indicating more faithful reconstruction of observed flight paths.

In the prediction module, the BiLSTM–AM model achieved average RMSE reductions of 57.66%, 80.35%, and 90.33% over a standard BiLSTM in the X, Y, and Z directions, respectively. These gains suggest the attention mechanism substantially improves three-dimensional trajectory forecasting accuracy over the baseline recurrent architecture.

Speed was another decisive advantage. The BiLSTM–AM model improved prediction speed by 85.68% relative to numerical integration, making the approach more suitable for real-time fire control applications where rapid trajectory estimates are required.

The study, authored by Shaoquan Wang, Qiang Li, and Chengming Li, was published on 25 October 2026 as an original paper. By combining reinforcement learning-based identification with attention-enhanced deep learning prediction, the framework offers an integrated alternative to traditional static identification and integration-heavy forecasting methods for projectile dynamics.

Subject of Research: Sequential deep reinforcement learning for projectile aerodynamic identification and attention-based BiLSTM trajectory prediction

Article Title: A Sequential DRL Framework for Projectile Aerodynamic Identification and an Attention-Based BiLSTM for Trajectory Prediction

Article References: Wang, S., Li, Q., & Li, C. (2026). A Sequential DRL Framework for Projectile Aerodynamic Identification and an Attention-Based BiLSTM for Trajectory Prediction. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01319-z

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01319-z

Keywords: reinforcement learning, TD3, drag coefficient, aerodynamic identification, trajectory prediction, BiLSTM, attention mechanism, Markov decision process, ballistics, fire control systems, RMSE, numerical integration