💻 computer science

Scientific Machine Learning-assisted Model Discovery from Telemetry Data

This paper introduces "Dyad Model Discovery," a semi-automated, engineer-in-the-loop workflow that uses scientific machine learning to identify missing physics in dynamic models by augmenting them with data-derived symbolic expressions, thereby improving the calibration and predictive accuracy of digital twins for HVAC and transportation refrigeration systems.

Original authors: Sebastian Micluta-Campeanu, Avinash Subramanian, Anas Abdelrehim, Ranjan Anantharaman, Rohit Dhumane, Brad Carman, Chris Rackauckas

Published 2026-03-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sebastian Micluta-Campeanu, Avinash Subramanian, Anas Abdelrehim, Ranjan Anantharaman, Rohit Dhumane, Brad Carman, Chris Rackauckas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a perfect digital twin of a refrigerated truck trailer. You want a computer simulation that predicts exactly how the temperature inside will change as the truck drives down the highway, stops at traffic lights, or faces a hot summer day.

To do this, engineers write a set of physics equations (like a recipe) describing how heat moves through the walls, how the air circulates, and how the refrigeration unit works. But here's the problem: real life is messy. Sometimes, the "recipe" is missing a secret ingredient, or the measurements aren't quite right. When you try to tune the simulation to match real-world data, it fails. The model just doesn't get it right.

This paper introduces a clever, semi-automated assistant called Dyad Model Discovery to fix this. Think of it as a "smart co-pilot" for engineers.

Here is how it works, broken down into simple steps:

1. The "Black Box" Fixer (Universal Differential Equations)

Imagine your physics model is a car engine that's running rough. You know the basic mechanics (pistons, fuel, spark plugs), but it's still sputtering.
Instead of throwing the engine away, the Dyad system attaches a neural network (a type of AI brain) to the engine. This AI brain acts like a "correction filter." It watches the engine run, sees where it's going wrong compared to real data, and whispers tiny adjustments to the engine's behavior to make it run smoothly.

The Analogy: It's like a GPS that knows your car's engine is slightly off. It doesn't rebuild the engine; it just gives you real-time steering corrections to keep you on the road.

2. The "Noise Filter" (Masking)

Once the AI brain has learned the corrections, it might be overzealous. It might be trying to fix things that don't actually need fixing, or it might be making tiny, unnecessary adjustments to 28 different parts of the engine.
The system then asks: "Which of these 28 adjustments actually matter?"
It looks at the data and realizes, "Hey, 26 of these corrections are basically zero. We only need to fix 2 specific things." It turns off the noise and keeps only the vital corrections.

The Analogy: Imagine a chef tasting a soup and adding salt, pepper, garlic, and 20 other spices. The AI tastes it again and says, "Actually, you only needed salt and pepper. Let's throw out the rest."

3. The "Detective Work" (Sensitivity Analysis)

Now that we know what to fix, we need to know what causes the problem. The system looks at the inputs (temperature, pressure, wall thickness) and asks: "Which of these inputs is actually driving the error?"
It creates a "heat map" (like a weather map showing hot spots) to see which variables are most sensitive. It might find that the length of the partition between the two cargo zones is the biggest culprit, while the color of the truck doesn't matter at all.

The Analogy: It's like a doctor diagnosing a patient. Instead of treating every symptom, the doctor identifies that the fever is caused by a specific infection, not by the patient's diet or sleep schedule.

4. The "Human Translator" (Symbolic Regression)

Here is the magic part. The AI has been making corrections using a complex, unreadable "black box" math formula. Engineers can't use that because they need to understand why the model is behaving that way.
The system takes the AI's complex math and translates it into simple, readable equations (like standard algebra). It turns the "black box" back into a "white box" that an engineer can read, understand, and trust.

The Analogy: The AI speaks "Robot," and the engineer speaks "Human." This step is the translator that turns the robot's complex code into a simple sentence like: "If the wall is thin, add extra cooling."

5. The "Engineer in the Loop"

The paper emphasizes that this isn't a "set it and forget it" magic button. The system acts as a suggestion engine.

It says: "I think we should change the equation for heat transfer. Here is my new formula. Do you agree?"
The human engineer looks at it, says "Yes, that makes sense," or "No, that doesn't fit our physics," and makes the final call.

Why Does This Matter?

In the real world, this helps companies like Trane Technologies (who make refrigeration units for trucks) build better digital twins.

Before: They had a model that was okay, but sometimes failed to predict temperatures accurately, especially when carrying different types of cargo (like frozen food in one zone and fresh produce in another).
After: Using this AI-assisted discovery, they found missing physics in their model. They added simple, new equations that the AI discovered from the data.
Result: The simulation became much more accurate (improving prediction by 3%, which is huge in this field), helping them plan routes and save energy.

In a nutshell: This paper describes a workflow where AI helps engineers find the missing pieces of their physics puzzles, but the human engineer remains the boss, deciding which AI suggestions to keep and turning complex math back into simple, understandable rules.

1. Problem Statement

The paper addresses the challenge of calibrating dynamic physical models for Transportation Refrigeration Units (TRUs), specifically the cargo box component.

The Core Issue: Traditional physics-based models often fail to calibrate to real-world telemetry data. This failure is frequently attributed to oversimplified assumptions or missing physics within the model equations, rather than just poor parameter tuning.
The Consequence: Inaccurate models lead to suboptimal fleet planning and route optimization, which is critical for energy efficiency and decarbonization in the refrigerated transport sector.
Specific Context: The authors focus on a two-zone cargo box model where heat transfer occurs between zones and with the ambient environment. The problem is complicated by multi-temperature requirements, significant parameter deviations across different units, and the difficulty of identifying latent dynamics from noisy telemetry data.

2. Methodology: Dyad Model Discovery

The authors propose a semi-automated workflow called Dyad Model Discovery, which integrates Universal Differential Equations (UDEs) with an "engineer-in-the-loop" approach. The process transforms a standard physics-based model into a hybrid model that learns missing physics from data.

The methodology consists of four sequential numerical stages:

A. Universal Differential Equations (UDE)

Concept: A neural network is appended to the time derivatives of the original system of Differential Algebraic Equations (DAEs).
Formulation: If the original system is $\frac{du}{dt} = f(u, x, p, t)$ , the UDE modifies it to:
$\frac{du}{dt} = f(u, x, p, t) + NN(u, x, p, t; \theta)$
where $NN$ is a neural network with weights $\theta$ that learns the residual (missing physics).
Training: The system is trained using telemetry data to minimize the residual between the model output and observed data.

B. Output Masking (Dimensionality Reduction)

Goal: To identify which specific state derivatives require correction, reducing the complexity of the neural network.
Process: After initial training, the magnitude of the neural network's output corrections is analyzed. Outputs with negligible magnitudes are "masked" (removed). The network is re-trained with only the top significant outputs (e.g., reducing from 28 outputs to 2).

C. Sensitivity Analysis (Input Reduction)

Goal: To determine which input variables most significantly influence the corrections.
Process: A Jacobian matrix is computed at the optimal parameters. The relative magnitudes of the columns indicate which state variables or parameters the corrections are most sensitive to. This reduces the input dimension of the neural network.

D. Symbolic Regression

Goal: To replace the "black box" neural network with interpretable, symbolic mathematical expressions.
Process: Using the reduced set of inputs and outputs identified in the previous steps, a symbolic regression algorithm (e.g., AI Feynman or similar) searches for mathematical expressions that map inputs to the correction terms.
Outcome: The result is a new set of symbolic equations that can be appended to the original physical model, making the "discovered" physics transparent and usable by engineers.

3. Key Contributions

Engineer-in-the-Loop Workflow: Unlike purely data-driven approaches (like SINDy) or fully automated black-box AI, this workflow provides suggestions to the engineer at every stage (architecture selection, masking decisions, variable selection). The engineer retains final approval to accept or reject changes.
First AI-Assisted Engineering Design Workflow: The authors claim this is the first implementation of an AI-assisted workflow specifically designed for engineering design and model refinement in this context.
Hybrid Physics-ML Integration: The method successfully bridges the gap between first-principles physics (Modelica/DAE models) and machine learning, using UDEs to capture latent dynamics that the original model missed.
Interpretability: By converting the neural network corrections into symbolic expressions, the method ensures the final model remains physically interpretable, unlike standard deep learning models.

4. Results

The method was tested on a digital twin of a two-zone cargo box using telemetry data from Trane Technologies.

Model Setup: The original model was a 36-state system (28 differential, 8 algebraic variables) derived from a low-to-medium fidelity Modelica model.
Performance Improvement:
- The UDE training achieved a 3% improvement in the loss function on unseen datasets compared to the baseline.
- Dimensionality Reduction: The initial neural network had 28 outputs. Sensitivity analysis revealed that only 2 outputs were necessary to achieve the same predictive performance, significantly simplifying the model.
- Symbolic Success: The symbolic regression successfully generated parametrized expressions for the corrections. These expressions depended on physical parameters (e.g., partition sizes, initial temperatures) and specific state variables (e.g., mass flow rates, wall temperatures).
Generalization: The resulting symbolic model was able to extrapolate to unseen configurations (different zone sizes and temperatures) that were not part of the training set.

5. Significance and Future Work

Impact: This approach offers a robust solution for improving the predictive accuracy of digital twins in HVAC&R and industrial systems without discarding existing physics-based knowledge. It enables better fleet planning and energy optimization.
Limitations & Future Directions:
- Dummy Derivatives: The current symbolic expressions sometimes include "dummy derivatives" (artifacts of the compilation process) rather than pure physical quantities. Future work aims to eliminate these.
- Pre-compilation Correction: Currently, corrections are applied post-structural simplification. Future work aims to modify the user's source code equations directly before compilation.
- Scalability: The workflow relies on the model being ported to the Dyad environment (JSML/Julia), which requires an initial integration effort.

In conclusion, the paper demonstrates that combining Universal Differential Equations with symbolic regression and human oversight creates a powerful tool for Model Discovery, allowing engineers to automatically identify and mathematically formalize missing physics in complex thermal-fluid systems.