💻 computer science

Conditional Distribution Estimation of Building Characteristics with Diffusion Models for Urban Energy Modeling

This paper proposes a conditional tabular diffusion model trained on 2.2 million buildings to generate realistic, complete building characteristics from partial data, thereby addressing data scarcity and improving the accuracy of urban energy modeling workflows.

Original authors: Saumya Sinha, Alexandre Cortiella, Rawad El Kontar, Andrew Glaws, Ryan King, Patrick Emami

Published 2026-04-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Saumya Sinha, Alexandre Cortiella, Rawad El Kontar, Andrew Glaws, Ryan King, Patrick Emami

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

The Big Problem: The "Half-Finished Puzzle"

Imagine you are trying to build a massive, incredibly detailed model of a city's energy usage. To do this accurately, you need a puzzle piece for every single house in the city. Each piece needs to tell you: How big is the house? What year was it built? What kind of heater does it have? How many people live there?

The problem is, for most houses, we only have a few pieces of the puzzle. We might know the address and the size, but we have no idea what kind of stove they use or when the roof was replaced. Without these missing pieces, our city model is blurry and inaccurate.

Usually, to fill these gaps, experts have to guess using "average" values (like assuming every house has a 1990s heater). But this is like saying every person in a city is 5'9" and likes pizza. It's not true, and it leads to bad planning.

The Solution: The "AI Detective" (Diffusion Models)

The authors of this paper built an AI detective called a Conditional Diffusion Model. Think of it as a super-smart artist who has studied millions of houses across the entire United States.

Instead of just guessing one "average" answer, this AI learns the patterns of how houses actually look. It understands that:

If a house is built in the 1920s, it's unlikely to have a modern heat pump.
If a house is very large, it probably has more than one bathroom.
If a house is in a cold climate, it likely has a specific type of furnace.

How It Works: The "Denoising" Process

The paper uses a technique called Diffusion, which is a bit like sculpting with fog.

The Foggy Start: Imagine you have a clear picture of a house, but then you slowly cover it in thick fog until you can't see anything. This is the "forward process."
The Reverse Process: Now, imagine you have a blank canvas covered in static noise (fog). The AI's job is to slowly wipe away the fog, step by step, to reveal a clear picture of a house.
The "Conditional" Twist: Here is the magic. The AI doesn't just guess randomly. You give it a few clues (the "conditions"). For example, you say, "This house is 2,000 square feet and was built in 1950." The AI then uses those clues to guide its "wiping away" process. It only generates house details that fit those clues.

If you ask it to fill in the "Heating System" for a 1950s house, it won't guess a solar panel (which is rare for that era). It will likely guess an oil or gas furnace, because that's what the patterns tell it.

What They Tested

The team trained this AI on a massive dataset called ResStock, which contains details on 2.2 million real US homes. They tested it in two ways:

The "Blind Test": They took real houses, hid some of the data (like the age or the heater type), and asked the AI to guess. They then compared the AI's guesses to the real hidden data. The AI was surprisingly good at guessing the distribution (the variety of answers), not just a single average.
The "Baltimore Case Study": They took a real neighborhood in Baltimore with 77 houses. They had some real data but were missing about 30% of the details. They fed the known data into the AI, let it fill in the blanks, and then ran a physics-based energy simulation (using a tool called URBANopt).
- The Result: The energy consumption predicted by the AI-filled data was almost identical to the energy consumption predicted by the "perfect" reference data. Even when they hid up to 10 different details at once, the AI kept the energy predictions accurate.

The Catch (and the Future)

The AI isn't perfect. Sometimes, for very specific or rare combinations of clues, it gets a little too confident and only gives one specific answer (like a "mode collapse"), rather than showing the full range of possibilities. But for the vast majority of cases, it works beautifully.

Why This Matters

This is a game-changer for city planners and energy companies.

Before: They had to spend years and millions of dollars collecting data, or they had to use bad guesses.
Now: They can use this AI to instantly generate realistic, complete profiles for millions of buildings.

In short: This paper teaches us how to use an AI artist to fill in the missing pieces of our city's energy puzzle, allowing us to plan for a greener, more efficient future without needing to know every single detail about every single house beforehand.

1. Problem Statement

Urban Building Energy Models (UBEMs) require detailed, building-level data (e.g., square footage, HVAC systems, occupant behavior, equipment vintage) to accurately simulate energy consumption. However, such granular data is often:

Unavailable or costly to acquire at scale.
Incomplete in existing large-scale datasets (e.g., ResStock), which often provide only regional (PUMA-level) statistics rather than individual building details.
Inadequate when filled with default prototype specifications, which fail to capture the diversity of real-world building stocks.

Current imputation methods often rely on deterministic point estimates (e.g., regression, mean imputation) or struggle with the heterogeneous nature of tabular data (mixing continuous and discrete/categorical variables). There is a need for a generative approach that can learn complex, multivariate conditional distributions to fill data gaps with realistic, diverse synthetic values.

2. Methodology

The authors propose a Conditional TabDDPM (Tabular Denoising Diffusion Probabilistic Model) framework designed to impute missing building characteristics conditioned on observed attributes.

Core Architecture

Base Model: Adapts the TabDDPM architecture, which uses separate diffusion processes for different data types:
- Gaussian Diffusion: Applied to continuous numerical features (e.g., square footage, electricity consumption).
- Multinomial Diffusion: Applied to categorical features (e.g., HVAC type, building vintage, appliance presence).
Conditional Generation: The model is trained to generate the "target" (missing) variables ( $x_{target}$ $x_{t a r g e t}$ ) conditioned on the "observed" variables ( $x_{cond}$ $x_{co n d}$ ).
- Dynamic Masking: During training, a random subset of features is masked for each row to simulate various missing-data scenarios. This forces the model to learn robust conditional dependencies regardless of which specific features are missing.
- Input Processing: Numerical features are quantile-transformed; categorical features are ordinal and one-hot encoded.
- Denoiser: A Multi-Layer Perceptron (MLP) predicts the noise (for numerical) and logits (for categorical) for the target variables at each diffusion timestep, given the noisy target and the clean observed conditions.

Training Strategy

Dataset: Trained on the ResStock dataset (2.2 million U.S. residential buildings), which includes 35 features (4 numerical, 31 categorical).
Loss Function: A weighted sum of Mean Squared Error (MSE) for numerical features and KL-divergence for categorical features.
Variants: Two model variants were trained:
1. Mixed-Imputation: Imputes both numerical and categorical features.
2. Categorical-Only: Imputes only categorical features while keeping all numerical features observed.

3. Key Contributions

Conditional Diffusion Framework: Development of a unified generative model capable of handling heterogeneous tabular data (discrete and continuous) to perform conditional imputation. Unlike deterministic models, it learns the full conditional probability distribution, allowing for the generation of diverse, plausible scenarios.
Scalability and Generalization: The model is trained once on a national-scale dataset (ResStock) and demonstrated to generalize to unseen geographical regions (Out-of-Distribution testing), avoiding the need to train separate models for every district.
End-to-End Energy Modeling Workflow: Integration of the generative model with URBANopt, a physics-based energy simulation tool. The study demonstrates a complete pipeline: from imputing missing building data to simulating energy load profiles.

4. Results and Evaluation

The model was evaluated using quantitative metrics, qualitative distribution analysis, and a real-world case study.

Quantitative Metrics

Distribution Fidelity: Measured using Jensen-Shannon (JS) distance for categorical variables and Wasserstein distance for numerical variables.
- The model achieved an average JS distance of 0.204 for categorical features (compared to 0.462 for a random baseline).
- Average Wasserstein distance for numerical features was 0.115.
Reconstruction Accuracy:
- RMSE (Numerical): 0.076 (normalized).
- Accuracy (Categorical): 0.759.
- The model significantly outperformed baselines (Mean/Mode imputation and Random guessing).

Qualitative Analysis

Univariate & Bivariate: The model successfully captured complex dependencies (e.g., the relationship between building vintage and HVAC type).
Out-of-Distribution (OOD): The model maintained performance on a test set from a different geographical region (Baltimore), demonstrating its ability to generalize patterns learned from the national dataset.
Limitations: The model exhibited "mode collapse" for a few specific variables (e.g., heating setpoint offset), where it produced a deterministic single output rather than a diverse distribution, though this did not significantly impact overall reconstruction accuracy.

Case Study: Baltimore Residential Region

Setup: 77 single-family attached buildings with partial data (some features known from OpenStreetMap/Zillow, others missing).
Task: Impute up to 10 missing categorical features (including critical ones like vintage and stories) and simulate energy loads.
Outcome:
- The simulated annual electricity consumption using the generated data remained within a ±10% error margin of the reference (curated) dataset, even with 10 features masked.
- Hourly load profiles matched the reference shape and magnitude closely, with minor underestimation only at peak hours when the maximum number of features were masked.

5. Significance

Accelerating Energy Modeling: This work provides a scalable solution to the "data gap" problem in urban energy modeling, reducing the reliance on manual data curation and expensive surveys.
Probabilistic vs. Deterministic: By generating a distribution of possible values rather than a single point estimate, the approach allows planners to explore multiple retrofit scenarios and uncertainty ranges, which is crucial for robust energy planning.
Generalizability: The ability to train a single global model and apply it to local contexts (OOD) makes this approach highly practical for widespread deployment in diverse communities.
Foundation for Future Work: The study establishes a pathway for integrating generative AI into physics-based simulation workflows, enabling more accurate, fine-grained, and data-driven energy management strategies.