skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling
Effective inclusion of physics-based knowledge into deep neural network models of dynamical sys- tems can greatly improve data efficiency and generalization. Such a priori knowledge might arise from physical principles (e.g., conservation laws) or from the system’s design (e.g., the Jacobian matrix of a robot), even if large portions of the system dynamics remain unknown. We develop a framework to learn dynamics models from trajectory data while incorporating a priori system knowledge as inductive bias. More specifically, the proposed framework uses physics-based side information to inform the structure of the neural network itself, and to place constraints on the values of the outputs and the internal states of the model. It represents the system’s vector field as a composition of known and unknown functions, the latter of which are parametrized by neural networks. The physics-informed constraints are enforced via the augmented Lagrangian method during the model’s training. We experimentally demonstrate the benefits of the proposed approach on a variety of dynamical systems – including a benchmark suite of robotics environments featur- ing large state spaces, non-linear dynamics, external forces, contact forces, and control inputs. By exploiting a priori system knowledge during training, the proposed approach learns to predict the system dynamics two orders of magnitude more accurately than a baseline approach that does not include prior knowledge, given the same training dataset.  more » « less
Award ID(s):
1646522
PAR ID:
10383170
Author(s) / Creator(s):
; ; ; ;
Editor(s):
Firoozi, R.; Mehr, N.; Yel, E.; Antonova, R.; Bohg, J.; Schwager, M.; Kochenderfer, M.
Date Published:
Journal Name:
4th Annual Conference on Learning for Dynamics and Control
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Learning for Dynamics and Control Conference)
    Matni, Nikolai; Morari, Manfred; Pappas, George J (Ed.)
    Many dynamical systems—from robots interacting with their surroundings to large-scale multi-physics systems—involve a number of interacting subsystems. Toward the objective of learning composite models of such systems from data, we present i) a framework for compositional neural networks, ii) algorithms to train these models, iii) a method to compose the learned models, iv) theoretical results that bound the error of the resulting composite models, and v) a method to learn the composition itself, when it is not known a priori. The end result is a modular approach to learning: neural network submodels are trained on trajectory data generated by relatively simple subsystems, and the dynamics of more complex composite systems are then predicted without requiring additional data generated by the composite systems themselves. We achieve this compositionality by representing the system of interest, as well as each of its subsystems, as a port-Hamiltonian neural network (PHNN)—a class of neural ordinary differential equations that uses the port-Hamiltonian systems formulation as inductive bias. We compose collections of PHNNs by using the system’s physics-informed interconnection structure, which may be known a priori, or may itself be learned from data. We demonstrate the novel capabilities of the proposed framework through numerical examples involving interacting spring-mass-damper systems. Models of these systems, which include nonlinear energy dissipation and control inputs, are learned independently. Accurate compositions are learned using an amount of training data that is negligible in comparison with that required to train a new model from scratch. Finally, we observe that the composite PHNNs enjoy properties of port-Hamiltonian systems, such as cyclo-passivity—a property that is useful for control purposes. 
    more » « less
  2. ; ; (, IEEE)
    Data-driven approaches are increasingly popular for identifying dynamical systems due to improved accuracy and availability of sensor data. However, relying solely on data for identification does not guarantee that the identified systems will maintain their physical properties or that the predicted models will generalize well. In this paper, we propose a novel method for data-driven system identification by integrating a neural network as the first-order derivative of the learned dynamics in a Taylor series instead of learning the dynamical function directly. In addition, for dynamical systems with known monotonic properties, our approach can ensure monotonicity by constraining the neural network derivative to be non-positive or non-negative to the respective inputs, resulting in Monotonic Taylor Neural Networks (MTNN). Such constraints are enforced by either a specialized neural network architecture or regularization in the loss function for training. The proposed method demonstrates better performance compared to methods without the physics-based monotonicity constraints when tested on experimental data from an HVAC system and a temperature control testbed. Furthermore, MTNN shows good performance in a control application of a model predictive controller for a nonlinear MIMO system, illustrating the practical application of our method. 
    more » « less
  3. ; ; ; ; (, Thirty-First International Joint Conference on Artificial Intelligence)
    Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance. 
    more » « less
  4. ; ; ; ; ; ; ; ; ; (, IEEE Transactions on Biomedical Engineering)
    This paper proposes a method to learn ap- proximations of missing Ordinary Differential Equations (ODEs) and states in physiological models where knowl- edge of the system’s relevant states and dynamics is in- complete. The proposed method augments known ODEs with neural networks (NN), then trains the hybrid ODE-NN model on a subset of available physiological measurements (i.e., states) to learn the NN parameters that approximate the unknown ODEs. Thus, this method can model an ap- proximation of the original partially specified system sub- ject to the constraints of known biophysics. This method also addresses the challenge of jointly estimating physio- logical states, NN parameters, and unknown initial condi- tions during training using recursive Bayesian estimation. We validate this method using two simulated physiolog- ical systems, where subsets of the ODEs are assumed to be unknown during the training and test processes. The proposed method almost perfectly tracks the ground truth in the case of a single missing ODE and state and performs well in other cases where more ODEs and states are missing. This performance is robust to input signal per- turbations and noisy measurements. A critical advantage of the proposed hybrid methodology over purely data-driven methods is the incorporation of the ODE structure in the model, which allows one to infer unobserved physiological states. The ability to flexibly approximate missing or inac- curate components in ODE models improves a significant modeling bottleneck without sacrificing interpretability. 
    more » « less
  5. ; ; (, PLOS Computational Biology)
    Alber, Mark (Ed.)
    Biological systems exhibit complex dynamics that differential equations can often adeptly represent. Ordinary differential equation models are widespread; until recently their construction has required extensive prior knowledge of the system. Machine learning methods offer alternative means of model construction: differential equation models can be learnt from data via model discovery using sparse identification of nonlinear dynamics (SINDy). However, SINDy struggles with realistic levels of biological noise and is limited in its ability to incorporate prior knowledge of the system. We propose a data-driven framework for model discovery and model selection using hybrid dynamical systems: partial models containing missing terms. Neural networks are used to approximate the unknown dynamics of a system, enabling the denoising of the data while simultaneously learning the latent dynamics. Simulations from the fitted neural network are then used to infer models using sparse regression. We show, via model selection, that model discovery using hybrid dynamical systems outperforms alternative approaches. We find it possible to infer models correctly up to high levels of biological noise of different types. We demonstrate the potential to learn models from sparse, noisy data in application to a canonical cell state transition using data derived from single-cell transcriptomics. Overall, this approach provides a practical framework for model discovery in biology in cases where data are noisy and sparse, of particular utility when the underlying biological mechanisms are partially but incompletely known. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email