In the field of simulation of dynamical systems, ordinary differential equations (ODE) are fundamental tools for modeling physical, biological or physiological processes. However, in real practice it often happens that we only know part of the equations that govern the system, while other sections remain hidden or are too complex to formulate analytically. In addition, the available measurements are often incomplete, as we only capture a subset of the state variables. This scenario poses an important challenge: how can we learn the missing dynamics and, at the same time, reconstruct the unobserved variables? A promising answer lies in combining machine learning techniques with known physical models, giving rise to so-called hybrid neural-physical models.
The classic approach of pure data-driven modeling—for example, neural networks trying to learn all the dynamics from scratch—tends to be opaque, requires large volumes of information, and often does not respect the underlying physical laws. On the other hand, a purely mechanistic model may be too restrictive or inaccurate if the omitted equations are relevant. The hybrid solution consists of keeping explicit the components of the system that we know (for example, conservation laws or linear relationships) and representing the unknown parts by means of a neural network. This not only improves accuracy, but also preserves the interpretability of the model, a critical aspect in sectors such as engineering or healthcare.
In this context, a particularly effective technique is guided learning using the Rauch-Tung-Striebel Smoother (RTS), a classical state estimation algorithm in dynamic systems with noise. The proposed method operates in two iterative stages. In the first stage, the parameters of the model are assumed (including the weights of the neural network that represents the missing dynamics) and the available measurements are used to infer the trajectories of the latent states using the RTS smoother. This step leverages the structure of the model and temporal information to optimally reconstruct the unmeasured variables. In the second stage, the smoothed trajectories are treated as labeled data, and backpropagation is employed to update the neural network weights, minimizing the error between the model prediction and the estimated trajectories. This cycle is repeated until a predefined convergence criterion is reached.
The power of this scheme lies in the fact that it combines Bayesian statistical inference (RTS) with deep learning, allowing the model to learn missing dynamics from partial and noisy measurements. In addition, it maintains the mechanistic structure where it is known, which facilitates the interpretation of the results and validation by experts. In tests carried out on reference systems – including linear, nonlinear and even rigid dynamics (stiff) – the method has demonstrated a remarkable ability to reconstruct latent states and make long-term predictions, outperforming purely data-based or exclusively mechanistic approaches.
From a business and technology perspective, these types of developments open up significant opportunities for industries where complex system modeling is essential. For example, in the energy industry to simulate electricity grids with renewable generation, in pharmacology to model the kinetics of drugs within the body, or in robotics to control systems with partially known dynamics. The ability to integrate physical knowledge with artificial intelligence allows for more robust solutions that are less dependent on large volumes of labeled data.
At Q2BSTUDIO, as a company specializing in software and technology development, we understand that every organization faces unique challenges in modeling its processes. For this reason, we offer artificial intelligence services for companies that include the implementation of hybrid neural-physical architectures, adapted to the specific needs of each client. Our team combines expertise in dynamic systems, machine learning, and custom application development to build platforms that integrate accurate predictive models, both on-premises and in the cloud.
The successful implementation of these models requires not only a solid theoretical knowledge, but also an adequate technological infrastructure. In this sense, the AWS and Azure cloud services we offer allow us to deploy scalable simulation systems, capable of processing large volumes of data and executing intensive neural network training. In addition, the incorporation of autonomous AI agents facilitates the continuous monitoring and adjustment of models in production, ensuring that they remain accurate in the face of changes in operating conditions.
Of course, data and model security is critical, especially when working with sensitive information or critical systems. Our cybersecurity department performs audits and penetration tests to protect both the algorithms and the infrastructures where they run. Likewise, the visualization and analysis of the results are fundamental for decision-making; therefore, we integrate business intelligence services with tools such as Power BI, which allow managers and analysts to explore predictions and simulate scenarios interactively.
The trend towards hybrid models that combine physics and machine learning is marking a before and after in systems engineering. Companies that traditionally relied on expensive experimental campaigns or simplified analytical models can now benefit from tailored software solutions that capture both expert knowledge and data complexity. If your organization faces the challenge of modeling systems with partially known dynamics, we invite you to explore how the combination of RTS smoothers and neural networks can transform your predictive capabilities. At Q2BSTUDIO we are prepared to accompany you at every stage of the process, from conceptualization to implementation and ongoing maintenance of these tools.
