PaperlayerView exact revision on arXiv

Source-linked AI summary

Physics-Informed Neural Nets for Control of Dynamical Systems

Eric Aislan Antonelo, Eduardo Camponogara, Laio Oriel Seman, Eduardo Rehbein de Souza, Jean P. Jordanou, Jomi F. Hubner

arXiv:2104.02556v3cs.LG

TL;DR

Conventional PINNs lack control inputs and degrade when simulating beyond their fixed training interval, limiting their use as predictive models in MPC. The paper introduces PINC, which conditions a PINN on initial state and control input and chains short-interval predictions. PINC supports arbitrary long-range simulation and reported simulation speedups of up to 30% on average, while the authors note unresolved model-mismatch and disturbance-rejection limitations.

  • Problem

    Conventional PINNs do not readily support optimal control or variable long-range simulation, despite the need for efficient predictive models when data are sparse or numerical simulation is costly.

  • Method

    PINC augments a continuous-time PINN with the system’s initial state and control input, then chains interval predictions for control-oriented simulation.

  • Results

    PINC supports longer-range intervals not fixed beforehand and speeds ODE simulation by up to 30% on average.

  • Takeaways & Limitations

    PINC makes physics-informed neural networks usable as predictive models in MPC while reducing the computational burden of differential-equation simulation.

  • Takeaways & Limitations

    PINC does not inherently handle modeling errors or completely reject disturbances, although control-level filtering can provide some robustness.

Abstract

from arXiv · show

Physics-informed neural networks (PINNs) impose known physical laws into the learning of deep neural networks, making sure they respect the physics of the process while decreasing the demand of labeled data. For systems represented by Ordinary Differential Equations (ODEs), the conventional PINN has a continuous time input variable and outputs the solution of the corresponding ODE. In their original form, PINNs do not allow control inputs, neither can they simulate for variable long-range intervals without serious degradation in their predictions. In this context, this work presents a new framework called Physics-Informed Neural Nets for Control (PINC), which proposes a novel PINN-based architecture that is amenable to control problems and able to simulate for longer-range time horizons that are not fixed beforehand, making it a very flexible framework when compared to traditional PINNs. Furthermore, this long-range time simulation of differential equations is faster than numerical methods since it relies only on signal propagation through the network, making it less computationally costly and, thus, a better alternative for simulation of models in Model Predictive Control. We showcase our proposal in the control of two nonlinear dynamic systems: the Van der Pol oscillator and the four-tank system.

1. Introduction

The paper identifies limitations in conventional PINNs for controlled and long-horizon dynamical-system simulation, then introduces PINC to address them. PINC adds control-relevant inputs, supports indefinite chained simulation, and targets faster MPC prediction.

  • 1. Introduction: The paper frames MPC as receding-horizon optimization that repeatedly uses a plant-prediction model to compute control actions.This motivates developing a physics-informed predictive model suitable for control loops.
  • 1. Introduction: PINC is a PINN-based architecture designed to make continuous-time physics-informed networks amenable to control problems.It is motivated by the absence of continuous-time PINN architectures readily supporting Multiple Shooting and MPC.
  • 1. Introduction: PINC augments continuous time with the system’s initial state and control input, learning an ODE solution conditioned on both over shorter intervals.This structure is inspired by multiple shooting and collocation methods.
  • 1. Introduction: Conventional PINNs degrade beyond the training interval, whereas PINC chains predictions by feeding each interval’s final state into the next interval.The proposed self-feedback scheme supports an indefinite horizon without significant prediction deterioration according to the authors.
  • 1. Introduction: PINC is intended to improve MPC real-time simulation because trained-network inference can replace numerical solution at each prediction-horizon timestep.The authors present this as better satisfying real-time requirements for differential-equation simulation.

2. Related Works

Prior work uses neural networks for MPC, long-time PINN integration, and control-oriented physical modeling, but the reviewed approaches differ in control inputs, discretization, or demonstrated control experiments. The paper positions PINC against these alternatives as a continuous-time control-oriented architecture.

  • 2. Related Works: Earlier MPC studies used neural networks as controllers, control-law approximators, or dynamical models, including Echo State Networks and feedforward networks.These approaches include imitation, direct control-cost training, trajectory linearization, and predictive-control calculations.
  • 2. Related Works: Parareal PINNs address long-time PDE integration by decomposing the problem into independently trained short-time PINNs supervised by a coarse solver.This differs from PINC’s chained prediction approach described elsewhere in the paper.
  • 2. Related Works: Related PINN-control proposals include a chaos-control study without actual control inputs or output control signals and a building model without demonstrated control experiments.The cited approaches also differ from this work in architecture or physical-loss treatment.
  • 2. Related Works: Physics-informed reinforcement learning learns state-transition dynamics using physical laws alongside real state-action samples and rewards.This represents a discrete transition-model formulation rather than the continuous-time PINC setup.

3.1. Physics-informed Neural Networks (PINNs)

A conventional PINN represents an ODE solution with a neural network taking continuous time as input and uses data and physics-based losses during training. The physics term penalizes differential-equation violations at collocation points, reducing reliance on measured training data.

  • 3.1. Physics-informed Neural Networks (PINNs): A PINN maps continuous time t to the dynamic-system state y and is trained to represent the solution of a nonlinear ODE.The network output is constrained by the ODE’s differential operator.
  • 3.1. Physics-informed Neural Networks (PINNs): PINN training combines a data-fitting mean squared error with the physics residual term, typically optimized using ADAM or L-BFGS.The data term uses boundary or initial-condition information, while the physics term regularizes the learned solution.
  • 3.1. Physics-informed Neural Networks (PINNs): The physics-informed loss penalizes mismatched behavior of y(t) at randomly sampled collocation points through the differential operator F(y).This term measures how closely the network adheres to the physical model.
  • 3.1. Physics-informed Neural Networks (PINNs): Experiments reported in the paper indicate that physics-informed training can drastically reduce the training-data size needed to learn dynamical behavior.The reduction is attributed to prior information incorporated through the physics residual.

3.2. Nonlinear Model Predictive Control

Nonlinear MPC predicts system behavior over a horizon and chooses control actions by solving a constrained optimization problem. Its formulation penalizes tracking error and control increments while enforcing system, inequality, and equality constraints.

  • 3.2. Nonlinear Model Predictive Control: At each control-loop instant, MPC compares model predictions with the real process and uses iterative constrained optimization to propose actions.This receding-horizon procedure motivates fast predictive models.
  • 3.2. Nonlinear Model Predictive Control: NMPC optimizes control actions over a prediction horizon using a nonlinear state-transition model and reference trajectory.The formulation focuses on systems whose model lacks linearity.
  • 3.2. Nonlinear Model Predictive Control: NMPC imposes state-transition, inequality, and equality constraints through the model functions f, h, and g.The optimization is solved at each time step, typically applying the first control increment.
  • 3.2. Nonlinear Model Predictive Control: The MPC objective penalizes quadratic reference-tracking error and control increments, weighted by diagonal matrices Q and R.The reference is defined over the penalized portion of the horizon.

3.3. Physics-Informed Neural nets-based Control (PINC)

PINC extends PINNs with initial-state and control inputs, dividing long simulations into shorter intervals that can be chained for MPC. It predicts interval endpoints through forward propagation, while physics-informed training can use sampled initial conditions and collocation points.

  • PINC architecture: PINC augments a continuous-time PINN with variable initial-state and control inputs, enabling conditioned ODE solutions over shorter intervals.The control is assumed constant within each interval t ∈[0, T ], while initial states and controls can vary across the full simulation.
  • PINC architecture: The same network solves successive intervals by fixing u[k], initializing from the previous state, and feeding its prediction back during free-run horizon simulation.Within MPC, predicted states are recursively reused during the prediction horizon, whereas the plant state can reset the network between control iterations.
  • Combining the intermediate solutions: A state at t = T is obtained by one forward propagation, avoiding intermediate numerical integration and providing the control interface used at each sampling period.The interface can provide ∂b_f^w/∂u for MPC solvers, and predicts y[k] at every Ts seconds.
  • Free-run simulation in the prediction horizon: MPC resets each prediction horizon to the plant’s true state, countering error accumulation that can occur when predicted states are recursively fed back during free runs.Error accumulation remains possible inside a future finite horizon because no real-process readings are available there.
  • Training: PINC training uses initial-condition data and physics collocation points, so an exact ODE model can replace measured process data for the training dataset.The training loss combines data-fit and physics terms, with collocation inputs sampling time, initial state, and control input.

3.4. Metrics

The evaluation uses generalization MSE for PINC predictions, alongside IAE and RMSE for control performance. The PINC prediction is compared with Runge-Kutta trajectories under the same control signal.

  • 3.4. Metrics: Generalization MSE is evaluated at discrete self-loop time steps by comparing PINC predictions with Runge-Kutta simulations under identical control inputs.The Runge-Kutta trajectory represents the true plant model.
  • 3.4. Metrics: IAE measures cumulative absolute tracking error across C control iterations and is especially suited to comparing runs with the same reference signal.
  • 3.4. Metrics: RMSE measures the average error behavior of the controller during evaluation.

3.5. PINC Algorithms

PINC training minimizes data and physics residual losses, while MPC uses the trained network as a predictive model to optimize control actions at each timestep.

  • 3.5. PINC Algorithms: PINC training minimizes the combined data and collocation losses using ADAM followed by L-BFGS optimization.
  • 3.5. PINC Algorithms: The MPC procedure optimizes a control input at each timestep using PINC predictions over specified control and prediction horizons.

4. Experiments

The experiments apply PINC to modeling and controlling the Van der Pol oscillator and four-tank system. The algorithms train a physics-informed network and embed it in MPC.

  • 4. Experiments: The experimental study evaluates PINC on the nonlinear Van der Pol oscillator and four-tank system.
  • 4. Experiments: The training algorithm computes gradients of combined data and physics losses from sampled data and collocation points, first using ADAM and then L-BFGS.
  • 4. Experiments: The training procedure saves the network with the lowest validation error after updating weights with both optimizers.
  • 4. Experiments: The MPC algorithm initializes predictions from the plant state, optimizes controls with the trained PINC model, and applies each resulting action to the plant.

4.1. Van der Pol Oscillator

The Van der Pol experiments assess PINC architecture, long-range simulation, and control. PINC supports variable control inputs and long-range self-loop prediction, with strong trajectory agreement and lower error than shorter-horizon conventional PINNs.

  • 4.1. Van der Pol Oscillator: The best reported generalization error is 10^-2.87 for a 10-layer network with 20 neurons per layer, while 40 data points are insufficient and Nf/Nt should exceed 4.
  • 4.1. Van der Pol Oscillator: PINC extends a 0.5-second training interval through chained self-loop predictions, whereas conventional PINNs require interval-specific training and degrade outside their trained horizon.Only the PINN trained for 10 seconds remains accurate through that interval in the reported comparison.
  • 4.1. Van der Pol Oscillator: At 10 seconds, conventional PINNs have higher RMSE than PINC when trained on shorter intervals, while the 10-second PINN is slightly better than PINC.The PINC disadvantage at 10 seconds is attributed to small self-loop error accumulation.
  • 4.1. Van der Pol Oscillator: PINC accommodates variable control inputs, unlike the fixed-input comparison used for the conventional PINN benchmark.
  • 4.1. Van der Pol Oscillator: The final network uses four hidden layers of 20 neurons, and extending training to K2 = 20,000 improves validation error by at least one order of magnitude.
  • 4.1. Van der Pol Oscillator: The PINC trajectory closely matches the target for a randomly generated 10-second control input while allowing predictions every 0.5 seconds.Intermediate trajectory values are obtained by varying continuous time while holding the initial state and control input fixed.

4.2. Four Tanks

The four-tank benchmark evaluates PINC on nonlinear multivariable control with coupled tank dynamics, constrained levels, and MPC-based actuation. PINC successfully controls the system while achieving nearly equivalent RMSE to an ODE/RK predictive model and reducing simulation time.

  • Four Tanks: The four-tank plant is a nonlinear multivariable benchmark with coupling and possible non-minimum-phase transmission zeros.Two pumps actuate four tanks, whose flows are governed by mass balances and a nonlinear Bernoulli orifice relation.
  • PINC Control: Five layers of 20 neurons are used because the plant has multiple inputs and outputs operating at different timescales.The reported training configuration uses Nt = 1,000, Nf = 100,000, K1 = 500, and K2 = 20,000.
  • PINC Control: PINC uses 5-step prediction and control horizons, corresponding to 50s, while resetting the predicted initial state to sensor measurements at each MPC optimization.The short horizon limits accumulated self-loop prediction error before the next state reset.
  • PINC Control: PINC successfully tracks the reference trajectories while respecting h3 and h4 constraints during the 2400s four-tank simulation.The controlled variables are h1 and h2, while h3 and h4 remain within [0.6, 5.5] cm.
  • PINC Control: 23.3% lower runtime: PINC averages 10.85s versus 14.15s for the four-tank ODE/RK model, with almost equivalent RMSE errors.IAE differs more between methods, while RMSE is reported as almost equivalent.
  • Sensitivity to Perturbations: Across 151 perturbation runs, IAE remained within an acceptable tolerance range despite random k1 and k2 deviations.Because PINC has no integrators, model mismatch can produce a small steady-state error that increases with parameter deviation.

4.3. Discussion

The discussion emphasizes PINC's runtime benefits after training while identifying training cost and equation-related learning difficulties as practical limitations. The approach can predict states directly over an interval without intermediate numerical integration.

  • Discussion: Up to 30% faster ODE simulation: trained PINC inference avoids numerical integration at intermediate points.The authors attribute possible larger gains to parallelized inference or extensions to PDEs, but those gains are prospective.
  • Discussion: Long training is a main obstacle, even though trained PINC networks can directly predict any state in [0, T].This avoids the intermediate-point integration required by numerical simulation methods.
  • Discussion: L-BFGS optimization is described as essential for obtaining a precise trained model, while skip connections may improve training of deeper networks.Skip connections are motivated by improved gradient propagation to deep layers.
  • Discussion: Discontinuous ODE operators and random neural-network initialization can create learning challenges, including invalid square-root arguments.The paper notes that fixes or workarounds may be applied in these cases.

5. Conclusion

PINC adapts PINNs for MPC and longer-range simulation without severe prediction degradation, while reducing computational burden after training. The conclusion identifies sparse additional data and broader differential-equation settings as future directions.

  • Conclusion: PINC makes PINNs amenable to MPC and supports longer-range intervals not fixed beforehand without severe prediction degradation.The framework is presented as opening application opportunities for control systems.
  • Conclusion: Using sparse additional data may make PINC-net training much faster than training from initial conditions alone.The paper reports using only initial conditions as real training data in the presented work.
  • Conclusion: Future work targets DAEs, PDEs, uncertain prior knowledge, and industrial control problems with noisy or sparse data.The authors expect computational savings to become more relevant as model complexity increases.
Loading 2104.02556v3…