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Physics-constrained Deep Learning of Multi-zone Building Thermal Dynamics
Jan Drgona, Aaron R. Tuor, Vikas Chandan, Draguna L. Vrabie
TL;DR
Building MPC requires thermal models that are accurate and computationally practical, yet existing white-, gray-, and black-box approaches face development, optimization, data, or generalization challenges. This paper addresses the gap with a structured recurrent neural model incorporating building-physics priors, inequality penalties, and Perron-Frobenius-based eigenvalue constraints. Using 10 days of data from a 20-zone office building, the approach generalizes across extended test periods and improves reported accuracy over prior state-of-the-art methods.
Problem
Building thermal model development limits practical MPC deployment because existing approaches trade off accuracy, computational efficiency, expert effort, data requirements, and out-of-distribution generalization.
Method
The method encodes building-physics structure in a recurrent neural architecture and adds penalty-based inequality constraints plus Perron-Frobenius-based eigenvalue bounds.
Results
Using 10 days of training data from a real-world 20-zone office building, the constrained structured model achieves 0.0052 test normalized open-loop MSE and outperforms reported state-of-the-art methods.
Takeaways & Limitations
Physics-constrained recurrent modeling can provide sample-efficient, physics-consistent building thermal dynamics models with reported accuracy and generalization benefits.
Takeaways & Limitations
The eigenvalue-based physical interpretation relies on relating building U-values to system eigenvalues, while accurate U-value information can be time-consuming to obtain.
Abstract
from arXiv · showhide
We present a physics-constrained control-oriented deep learning method for modeling building thermal dynamics. The proposed method is based on the systematic encoding of physics-based prior knowledge into a structured recurrent neural architecture. Specifically, our method incorporates structural priors from traditional physics-based building modeling into the neural network thermal dynamics model structure. Further, we leverage penalty methods to provide inequality constraints, thereby bounding predictions within physically realistic and safe operating ranges. Observing that stable eigenvalues accurately characterize the dissipativeness of the system, we additionally use a constrained matrix parameterization based on the Perron-Frobenius theorem to bound the dominant eigenvalues of the building thermal model parameter matrices. We demonstrate the proposed data-driven modeling approach's effectiveness and physical interpretability on a dataset obtained from a real-world office building with 20 thermal zones. Using only 10 days' measurements for training, we demonstrate generalization over 20 consecutive days, significantly improving the accuracy compared to prior state-of-the-art results reported in the literature.
1. Introduction
Building MPC depends on thermal models that balance accuracy, computational efficiency, data requirements, and physical reliability. The paper proposes structured, physics-constrained deep learning to reduce modeling effort while improving generalization and interpretability.
- Motivation: Building thermal model development is the main practical bottleneck for deploying model predictive control.Thermal dynamics are high-dimensional, nonlinear, and potentially costly or nondifferentiable for gradient-based optimization.
- Motivation: Purely black-box models can be computationally efficient but often require large datasets and may generalize poorly beyond training conditions.Gray-box alternatives remain difficult to identify because they involve challenging non-convex optimization problems.
- Proposed approach: The proposed constrained deep-learning paradigm encodes structural assumptions and constraints into a generic neural architecture without manual redesign for each building.This approach is presented as an alternative to classical white-, gray-, and black-box modeling approaches.
- Evaluation: The paper demonstrates physics-constrained recurrent models on a real-world 20-zone office-building dataset using only 10 days of training data.The reported evaluation emphasizes accuracy, generalization, data efficiency, and physical interpretability.
- Proposed approach: The method combines block-structured recurrent dynamics, Perron-Frobenius-based eigenvalue constraints, penalty methods, and multi-step multi-term training losses.These components target structural fidelity, stability and dissipativity, physically meaningful bounds, and long-term prediction behavior.
- Contribution: The authors describe the combined structured recurrent and physics-inspired constrained architecture as novel for real-world building thermal dynamics modeling.The stated contribution concerns the combination of these elements rather than any single modeling component.
2. Methods
The method builds a control-oriented recurrent model by embedding building-thermal structure, stable eigenvalue parameterization, and penalty-based physical bounds into neural dynamics. It uses multi-step training objectives to encourage accurate and physically plausible predictions.
- Building Thermal Dynamics: The thermal model represents envelope states, zone-temperature measurements, weather and occupancy disturbances, and HVAC heat flows within a partially observable dynamical system.The underlying model separates spatial heat transfer, HVAC-driven temperature increments, and nonlinear weather or occupancy effects.
- Structured Recurrent Neural Dynamics Model: The architecture encodes block-structured state-space dynamics with separate neural components for state, input, and disturbance effects.A state observer maps a past N-step observation window to initial latent conditions for partially observable systems.
- Eigenvalue Constraints: Stable eigenvalues encode the dissipative character of building thermal dynamics, while Perron-Frobenius-based parameterization bounds the dominant eigenvalues of the learned transition matrix.The parameterization combines a damping matrix with row-wise-softmax weights to form a state transition matrix bounded by λmin and λmax.
- Inequality Constraints via Penalty Methods: Penalty functions impose inequality constraints that keep model variables within physically realistic lower and upper bounds.Slack variables measure violations and are heavily weighted in the training objective; ReLU functions implement the penalties.
- Training Objective: The recurrent model is trained over an N-step prediction window using mean squared prediction error together with regularization and state-difference penalties.The state-difference term promotes smoother and more physically plausible state trajectories, while multi-step optimization improves accuracy and generalization.
3. Experimental Case Study
The study evaluates physics-constrained recurrent neural models for a 20-zone commercial building using limited time-series data, comparing structure, penalties, architectures, and eigenvalue constraints. The constrained structured model achieves the strongest reported performance while improving physical interpretability and long-horizon behavior.
- Dataset and setup: The dataset represents 20 zone-temperature outputs, 40 HVAC inputs, and one ambient-temperature disturbance sampled every 15 minutes over 30 days.The data are split evenly into training, validation, and test sets, with 960 datapoints in each.
- Dataset and setup: The experiments compare structured and unstructured recurrent models with or without penalty and eigenvalue constraints.The evaluation covers open-loop and N-step simulation performance on held-out data.
- Model performance: 0.0052 test open-loop MSE is achieved by the constrained structured model, corresponding to roughly 0.18K error per output.The same model scores 0.0091 on development data and 0.0143 on training data, and generalizes across 30 days after training on 10 days.
- Model performance: Longer prediction horizons improve open-loop performance for constrained models, whereas unconstrained models deteriorate beyond N = 16.A horizon of N = 64 also minimizes the gap between N-step loss and open-loop performance.
- Physics-inspired structure: Combining physics-inspired structure and constraints reduces error by 15% versus unstructured and unconstrained models.Separate blocks represent state, HVAC-input, and disturbance dynamics, supporting interpretable physical roles.
- Neural block architecture: RNN and MLP blocks scale better than ResNet blocks, with RNNs favoring shorter horizons and MLPs performing better at N = 64.The comparison concerns open-loop and N-step MSE across structured and unstructured models.
- Eigenvalue constraints: Perron-Frobenius eigenvalue constraints improve structured-model performance at shorter horizons but provide no benefit to unstructured models.The constraints encode an inductive bias toward dissipative building-envelope dynamics that are represented specifically by the structured state map.
- Eigenvalue constraints: The constrained state-map eigenvalues remain between λmin = 0.8 and λmax = 1.0, bounding learned dissipativeness within prescribed values.Only two dominant dynamical modes exceed 0.8, one per state-map layer.
4. Conclusions
The study presents a sample-efficient, physics-consistent deep learning method for building thermal dynamics that improves prior state-of-the-art modeling results using measurements from a real-world office building.
- 4. Conclusions: Using only 10 days’ measurements, the method significantly improves prior state-of-the-art results on a real-world large-scale office building modeling task.The improvement is attributed to structural assumptions, eigenvalue constraints on weights, and penalty methods applied to neural-network outputs.