A Reduced Order Model (ROM) is a simplified version of a complex computational model or simulation, designed to provide results much faster and with fewer computational resources. A ROM is similar to a subway map in this analogy. It removes extraneous details, such such as individual streets or buildings, to highlight only the most pertinent information—like subway lines and stations—for a specific purpose, which is to facilitate rapid movement between points. This simplification allows engineers and scientists to understand the dominant behaviors of a system without the burden of full complexity.
The Need for Simplification in Complex Systems
The counterpart to a reduced order model is the full-order model (FOM). These FOMs are built to capture every detail and interaction within a system, offering a high degree of accuracy. However, this precision comes at a substantial computational expense. Simulating complex phenomena, such as intricate weather patterns, the flow of air over an aircraft wing, or the detailed behavior of a new material, can demand days, weeks, or even months of processing time on powerful supercomputers.
This intense computational burden is often exacerbated by what scientists refer to as the “curse of dimensionality.” As the number of variables or parameters in a system increases, the computational resources needed to model its behavior can grow exponentially. This exponential growth makes many full-order simulations impractical for scenarios that require quick iterations, such as optimizing a design or controlling a system in real time.
How a Reduced Order Model is Built
Creating a reduced order model generally involves a systematic process that distills the complex behavior of a system into a more manageable form. This process often begins with data generation, where the expensive full-order model is run multiple times under various conditions. Alternatively, data can be collected from physical experiments, providing a rich dataset that captures the system’s responses.
After collecting this extensive data, algorithms analyze it to identify the most significant patterns or “modes” that define the system’s overall behavior. This step, known as dimensionality reduction, is where the “reduction” occurs. A widely used technique for this is Proper Orthogonal Decomposition (POD), which conceptually operates like a composer identifying the few core melodies or themes that characterize a complex composition. POD helps in finding a smaller set of basis functions that can approximate the high-dimensional data efficiently.
A new, simpler mathematical model is then constructed using only these identified dominant patterns. This new model can approximate the behavior of the full system with significantly fewer calculations. By focusing on the most influential components and discarding less impactful details, the reduced order model can deliver insights into system behavior in a fraction of the time.
Projection-Based Versus Data-Driven Methods
Reduced order models are categorized into two main families: projection-based methods and data-driven methods. Projection-based ROMs begin with the original, detailed physics equations that govern the full system. They then use mathematical techniques, such as Galerkin projection, to directly simplify these equations onto a lower-dimensional subspace. This approach “simplifies the rulebook,” maintaining a strong connection to the underlying physical laws.
In contrast, data-driven ROMs operate like a “black box,” learning system behavior directly from observed data without relying on explicit physics equations. These methods employ machine learning and artificial intelligence algorithms to identify patterns and relationships within the collected simulation or experimental data. This approach, akin to “learning the behavior, not the rules,” has gained popularity with advancements in AI, particularly for systems where the governing equations are either unknown or too complex to simplify directly.
Applications in Science and Engineering
Reduced order models are transforming how scientists and engineers approach complex problems across numerous fields. In aerospace engineering, ROMs enable rapid design iterations, allowing engineers to quickly test thousands of wing variations to identify the most aerodynamically efficient or fuel-saving shapes. They can predict surface pressure distributions and static aeroelastic loads, which aid in optimizing aircraft designs.
The automotive industry benefits from ROMs by simulating vehicle crash tests in near real-time. This allows for faster iteration on safety designs, reducing the need for expensive physical prototypes. Similarly, in medical science, ROMs facilitate the creation of “digital twins” of patient-specific organs, such as hearts or arteries. These digital replicas can simulate blood flow and predict the effectiveness of medical interventions before they are performed.
ROMs also find application in environmental sciences, particularly in weather and climate modeling. By speeding up simulations, they allow for faster uncertainty quantification in weather forecasting. This provides a more comprehensive understanding of potential weather outcomes and improves forecast reliability.