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Deep State Space Models for Nonlinear System Identification
Daniel Gedon, Niklas Wahlström, Thomas B. Schön, Lennart Ljung
TL;DR
Nonlinear system identification can benefit from flexible black-box models, while deep state-space models also represent output uncertainty. This paper explains and benchmarks six deep SSMs, finding them competitive with classic identification methods and reporting conservative uncertainty estimates.
Problem
The paper addresses how deep-learning models can extend nonlinear system identification, where established methods include parametric models and deep learning offers black-box modeling for complex systems.
Source: §1 · p. 1Method
The paper explains deep SSM structure and variational parameter learning, then evaluates six models in a unified implementation on three identification problems.
Source: §1 · p. 1, §2.3 · p. 2Results
Deep SSMs are competitive with classic identification methods, and their uncertainty estimates appear as conservative as established uncertainty-quantification methods.
Source: §5 · pp. 5–6Takeaways & Limitations
Deep SSMs extend the nonlinear-identification toolbox with a probabilistic black-box model class that estimates uncertainty in system dynamics.
Source: §1 · p. 1, §5 · p. 5The study covers only a variational-inference subclass of deep SSMs and calls for broader model classes and more nonlinear identification benchmarks.
Source: §5 · p. 6
Abstract
from arXiv · showhide
Deep state space models (SSMs) are an actively researched model class for temporal models developed in the deep learning community which have a close connection to classic SSMs. The use of deep SSMs as a black-box identification model can describe a wide range of dynamics due to the flexibility of deep neural networks. Additionally, the probabilistic nature of the model class allows the uncertainty of the system to be modelled. In this work a deep SSM class and its parameter learning algorithm are explained in an effort to extend the toolbox of nonlinear identification methods with a deep learning based method. Six recent deep SSMs are evaluated in a first unified implementation on nonlinear system identification benchmarks.
1. INTRODUCTION
Deep SSMs offer a flexible, probabilistic deep-learning approach to nonlinear system identification, connecting deep learning with automatic control. The paper develops this model class and its learning algorithm and compares six models in a unified implementation.
1. INTRODUCTION
- Parametric state-space models can encode pre-existing system structure and yield precise identification, while a wide range of identification algorithms is available.The paper contrasts this established system-identification toolkit with deep learning's use as a black-box modeling approach.Source: §1 · p. 1
- Deep SSMs use neural networks for flexibility, recurrent hidden states for temporal data, and probabilistic outputs to capture uncertainty.The paper notes prior uses of these advantages in handwritten-text and speech generation, where nonlinear dynamics and uncertainty quantification matter.Source: §1 · p. 1
- Deep SSMs are introduced as a deep-learning model class for nonlinear system identification, extending the identification toolbox and bringing system identification and deep learning closer.The paper distinguishes model structure from parameter estimation and compares six deep SSMs in one implementation.Source: §1 · p. 1
2. DEEP STATE SPACE MODELS FOR SEQUENTIAL DATA
The paper builds deep state-space models by combining recurrent neural networks with variational autoencoders. Their recurrent latent dynamics and probabilistic outputs provide the basis for the model variants studied.
2. DEEP STATE SPACE MODELS FOR SEQUENTIAL DATA
- A deep SSM uses a latent state to carry past information, with deep neural mappings defining state updates and output predictions.The paper frames the model around observed input-output sequences and predicted outputs.Source: §2 · p. 2
2.1 Recurrent Neural Networks
- GRUs are used in this paper for recurrent sequence modeling because of their structural simplicity.The recurrent hidden state is updated from the previous state and current input; the paper notes that GRUs and LSTMs yield empirically similar results.Source: §2.1 · p. 2
2.2 Variational Autoencoders
- A VAE maps data into a low-dimensional latent variable through an encoder and uses a decoder to generate data of approximately the same distribution.Because the true posterior is generally intractable, the encoder supplies a parametric Gaussian approximation.Source: §2.2 · p. 2
2.3 Combining RNNs and VAEs into deep SSMs
- The VAE-RNN stacks a VAE on an RNN, with recurrently updated prior parameters defining latent-state transitions and a separate generative network producing outputs.The paper also studies VRNN, VRNN-I, STORN, and Gaussian-mixture-output variants; VRNN additionally feeds the previous latent variable into the recurrence, while VRNN-I uses a static prior.Source: §2.3 · pp. 2–3
3. MODEL PARAMETER LEARNING
The learning method replaces an intractable likelihood objective with a variational lower bound. For deep SSMs, the objective extends across time, with the VAE-RNN factorization avoiding a smoothing step.
3.1 Cost Function for the VAE
- The VAE is trained by maximizing an evidence lower bound (ELBO) rather than the intractable log-likelihood.Its reconstruction term encourages data reconstruction, while the KL-divergence term regularizes the approximate posterior toward the prior.Source: §3.1 · p. 3
3.2 Cost Function for Deep SSMs
- For a generic deep SSM, the ELBO sums contributions over time, and a posterior depending on all time-step inputs and outputs requires smoothing.In the VAE-RNN, the factorization makes latent states independent given recurrent states, so no smoothing step is needed.Source: §3.2 · p. 3
4. NUMERICAL EXPERIMENTS
Six deep SSMs are evaluated on a linear Gaussian toy system and two nonlinear benchmarks using held-out data and open-loop comparisons. Results show competitive identification, with performance depending on model, training-data volume, and benchmark.
4. NUMERICAL EXPERIMENTS
- The experiments evaluate six models using training and validation data for learning and selection, with a separate test set for final performance.Evaluation uses open-loop output sequences; RMSE measures output error, while NLL quantifies how likely the true data point is under the model output distribution.Source: §4 · pp. 3–4
4.1 Toy Problem: Linear Gaussian System
- Deep SSMs approach state-of-the-art accuracy on the linear Gaussian toy problem, and more complex deep SSMs yield more accurate results.SSEST assumes a linear model, whereas the deep SSMs fit flexible nonlinear models.Source: §4.1 · p. 4, Table 1 · p. 4
- STORN captures uncertainty on the toy problem, but its estimated bounds are conservatively wider than the true uncertainty.The open-loop comparison shows the test-data results for STORN and SSEST, with shaded areas depicting uncertainty.Source: §4.1 · p. 4, Figure 3 · p. 4
4.2 Narendra-Li Benchmark
- STORN outperforms the other Narendra-Li models, whose performance is similar; additional training data improves RMSE and NLL until results plateau.The authors attribute STORN's advantage to its second recurrent inference network, which allows more accurate learned state representations.Source: §4.2 · p. 4, Figure 4 · p. 5
- On Narendra-Li, STORN reaches RMSE values close to gray-box models, but deep SSMs require more training samples than classic methods.The literature methods compared in Table 2 do not estimate uncertainty, so NLL is unavailable for those comparisons.Source: §4.2 · p. 5, Table 2 · p. 5
- For the Narendra-Li open-loop test, STORN identifies the nonlinear dynamics well but produces uncertainty bounds more conservative than the true bounds.Fig. 5 compares the identified STORN model with the true data and shows mean values and ±3 standard deviations.Source: §4.2 · p. 5, Figure 5 · p. 5
4.3 Wiener-Hammerstein Process Noise Benchmark
- STORN yields the best results on the Wiener-Hammerstein benchmark, while even the simple VAE-RNN identifies this complex benchmark well.Increasing hidden-state dimension has a rather limited influence; larger networks generally tend to produce more accurate results.Source: §4.3 · p. 5
5. CONCLUSION AND FUTURE WORK
The paper introduces deep state space models and variational-inference-based learning, evaluates six models on three identification problems, and reports that the class is competitive with classic identification methods. Its conclusion notes that the study covers a variational-inference subclass and calls for broader models and benchmarks; Appendix A presents graphical models for the studied variants.
5. CONCLUSION AND FUTURE WORK
- Deep state space models were evaluated on three system-identification problems, with results indicating competitiveness against classic identification methods.The paper presents the model class as an extension to the nonlinear-identification toolbox.Source: §5 · p. 5
- The probabilistic models estimate uncertainty in the dynamics, with behavior described as as conservative as established uncertainty-quantification methods.The authors relate this conservative behavior to existing variational-inference literature.Source: §5 · p. 6
- The study is limited to deep SSMs based on variational inference; the authors call for broader model classes and more nonlinear benchmarks.They also identify automatic-control applications and use of latent states as future research directions.Source: §5 · p. 6
Appendix A. GRAPHICAL MODELS FOR DEEP SSMS
- Appendix A presents graphical models for the other studied deep SSMs, following the paper's focus on VAE-RNN.The supplied WH benchmark visuals identify RMSE results for varying hidden dimension, but provide no cell values here.Source: Figure 6 · p. 6, Table 3 · p. 6, Appendix A · p. 8
A.1 VRNN
The VRNN uses a recurrent structure that incorporates the previous latent variable, while its generative network lets the hidden state directly influence the output. Its Gaussian and Gaussian-mixture variants share the network architecture but use different output distributions.
A.1 VRNN
- The VRNN recurrence uses the previous latent variable, and its generative network lets the hidden state directly influence the output.Both features provide more flexibility for the network output distribution.Source: §A.1 · p. 8
- VRNN-Gauss and VRNN-GMM share a network architecture but differ in whether the output distribution is Gaussian or a Gaussian mixture.The VRNN graphical model is shown in Fig. A.1.Source: §A.1 · p. 8, Figure A.1 · p. 8
- The appendix presents factorizations of the VRNN's joint true posterior and joint approximate posterior.The supplied factorization expressions are partial.Source: §A.1 · p. 8
A.2 VRNN-I
VRNN-I modifies VRNN by using a static prior in the generative network rather than a prior that changes over time with the recurrent network.
A.2 VRNN-I
- VRNN-I uses a static generative-network prior instead of one that changes temporally through the recurrent network.Its graphical model is shown in Fig. A.2.Source: §A.2 · p. 8, Figure A.2 · p. 8
- VRNN-Gauss-I and VRNN-GMM-I differ in output distribution, not network structure.The true and approximate joint posterior distributions otherwise follow the VRNN formulation, with the true posterior using a static prior.Source: §A.2 · p. 8
A.3 STORN
STORN adds a forward-running recurrent network to the inference network, and uses a static prior in its generative network. The added recurrence is described as helping encode the output distribution more precisely.
A.3 STORN
- STORN adds a forward-running recurrent network to the inference network, which helps encode the output distribution more precisely.The added recurrence is a GRU with hidden dimension ddim, matching the other recurrence's hdim.Source: §A.3 · p. 9
- STORN uses a static prior in its generative network, as does VRNN-I.Its graphical model is shown in Fig. A.3.Source: §A.3 · p. 9, Figure A.3 · p. 9
- The appendix also presents a factorization of STORN's joint approximate posterior.The supplied factorization expression is partial.Source: §A.3 · p. 9
Appendix B. TOY PROBLEM: LINEAR GAUSSIAN SYSTEM
The toy-system evaluation averages identifications over repeated realizations and compares uncertainty estimates against the true bounds and SSEST.
Appendix B. TOY PROBLEM: LINEAR GAUSSIAN SYSTEM
- Results are averaged over 50 Monte Carlo identifications, with new training and validation sequences drawn for each identification.The sequences share the same statistical properties.Source: Appendix B · p. 9
- All identified systems use the same test data set to make performance measures comparable.Source: Appendix B · p. 9
- STORN’s uncertainty quantification is comparable to SSEST’s, although no hyperparameter fine-tuning was performed.The reported uncertainty estimates are conservative relative to the system’s true uncertainty bounds.Source: Appendix B · p. 9
Appendix C. NARENDRA-LI BENCHMARK
The appendix describes the Narendra–Li benchmark and its excitation and measurement-noise setup.
Appendix C. NARENDRA-LI BENCHMARK
- The benchmark’s true dynamics are specified by a second-order model.Source: Appendix C · p. 9
- Additional measurement noise, e_k ∼ N(0, 0.1), is added to the original problem to make it more challenging.Source: Appendix C · p. 10
- Training and validation inputs use uniform random noise in [−2.5; 2.5], while the test input is defined by a separate sequence.The passage begins specifying the test sequence but does not provide its complete form.Source: Appendix C · p. 10
Appendix D. WIENER-HAMMERSTEIN PROCESS NOISE BENCHMARK
The Wiener–Hammerstein process-noise benchmark evaluates the deep SSMs on two test sets, with reported time evaluations indicating accurate identification and conservative uncertainty bounds.
Appendix D. WIENER-HAMMERSTEIN PROCESS NOISE BENCHMARK
- Identifications using available data sets with more samples yield tighter uncertainty bounds, but are omitted to preserve comparability with literature methods.Source: Appendix D · p. 10
- Comparison methods use 8192 training samples each, except PNLSS, which uses 9 realizations of 8192 samples each.Source: Appendix D · p. 10
- The time evaluations indicate accurate identification of the complex dynamics on both test data sets, with uncertainty bounds similarly conservative to those in the Narendra–Li benchmark.The evaluations show the best Tab. 3 results: STORN with hdim = 40, zdim = 3, and nlayers = 3.Source: Appendix D · p. 10, Figure D.1 · p. 11