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Data-driven discovery and control of multistable nonlinear systems and hysteresis via structured Neural ODEs

Ike Griss Salas, Ethan King

Abstract

Many engineered physical processes exhibit nonlinear but asymptotically stable dynamics that converge to a finite set of equilibria determined by control inputs. Identifying such systems from data is challenging: stable dynamics provide limited excitation and model discovery is often non-unique. We propose a minimally structured Neural Ordinary Differential Equation (NODE) architecture that enforces trajectory stability and provides a tractable parameterization for multistable systems, by learning a vector field in the form $F(x,u) = f(x)\,(x - g(x,u))$, where $f(x) < 0$ elementwise ensures contraction and $g(x,u)$ determines the multi-attractor locations. Across several nonlinear benchmarks, the proposed structure is efficient on short time horizon training, captures multiple basins of attraction, and enables efficient gradient-based feedback control through the implicit equilibrium map $g$.

Data-driven discovery and control of multistable nonlinear systems and hysteresis via structured Neural ODEs

Abstract

Many engineered physical processes exhibit nonlinear but asymptotically stable dynamics that converge to a finite set of equilibria determined by control inputs. Identifying such systems from data is challenging: stable dynamics provide limited excitation and model discovery is often non-unique. We propose a minimally structured Neural Ordinary Differential Equation (NODE) architecture that enforces trajectory stability and provides a tractable parameterization for multistable systems, by learning a vector field in the form , where elementwise ensures contraction and determines the multi-attractor locations. Across several nonlinear benchmarks, the proposed structure is efficient on short time horizon training, captures multiple basins of attraction, and enables efficient gradient-based feedback control through the implicit equilibrium map .

Paper Structure

This paper contains 21 sections, 3 theorems, 59 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Preliminaries
  4. Methodology
  5. System identification
  6. Control
  7. Continuous feedback control
  8. Constrained feedback control
  9. Theoretical analysis of gradient-based control
  10. One dimensional systems
  11. Gradient descent
  12. Continuous-time gradient flow
  13. Rank-deficient case
  14. Numerical results
  15. Mixing tanks
  16. ...and 6 more sections

Key Result

Lemma 1

Let $x^* \in \mathbb{R}$ be a fixed point of the scalar system for a given constant input $u \in \mathcal{U}$ (i.e. $x^* = g(x^*,u)$). Suppose $f\in C^1(\mathbb{R})$ with $f(x^*)<0$, and the partial map $g(\cdot, u) \in C^1(\mathbb{R})$. Then $x^*$ is a locally exponentially stable equilibrium if and only if

Figures (19)

  • Figure 1: Top: Simulated steady states at $t = 1000$ for control settings $v,p \in [0,1]$. Red triangles indicate training configurations $v,p = 0.1 + 0.1 i$ for $i = 0,\dots,8$. Bottom: 21 training trajectories for the configuration $p = 0.5$ and $v = 0.25$. Vertical line denotes training cutoff, $t=200$.
  • Figure 2: Six trajectories with initial conditions $\boldsymbol{x}(0)$ = ($0.1$, $0.1$), ($0.9$, $0.2$), ($0.2$, $0.9$), ($0.1$, $0.6$), ($0.6$, $0.1$), and ($0.8$, $0.8$) for 25 control configurations plotted on true phase portraits generated from \ref{['eqn:two-tanks']} (black line) compared against model simulated trajectories from $f_\theta(\boldsymbol{x})(\boldsymbol{x}-g_\theta(\boldsymbol{x}, \boldsymbol{u}))$ (blue dashed line). True and model steady states are denoted by black and red crosses, respectively.
  • Figure 3: Absolute error, for tanks 1 and 2, between model and true steady states for control values $p,v \in [0,1]$. Model steady states are obtained from solving associated root finding problem, $\boldsymbol{x} = g_\theta(\boldsymbol{x}, \boldsymbol{u})$. True steady states obtained from solving system numerically at $t=1000$.
  • Figure 4: Stochastic feedback control experiments with $10$ randomized targets reassigned every $t=500$. Top: Itô simulations with state-proportional noise of scale $\sigma=0.01$. $g_\theta$ for both tanks 1 and 2 plotted, where control iteration and strength are $k=10$ and $\eta=0.1$, respectively. Bottom: Constrained continuous feedback control solving \ref{['eqn:constrained-control']} jointly with the stochastic system where control constraint is given by \ref{['eqn:tank-control-constraint']}.
  • Figure 5: Simulated trajectories (with $t$ on a logscale) for model $f_\theta(x)(x-g_\theta(x,\lambda))$ (blue dashed line) trained on data up to $t=0.25$ (red dashed line) and true dynamics given by \ref{['eqn:sym-hyst']} (black solid lines) for control values $\lambda = [-1,-0.55,-0.45,-0.4,-0.35,0.35,0.4,0.45,0.55,1]$ up to $t=100$ with bistable cases highlighted in red.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Lemma 1: Linear stability condition
  • proof
  • Theorem 1: Local contraction of $g$
  • proof
  • Corollary 1: Geometric convergence of the iteration \ref{['eqn:g-iter']}
  • proof
  • Remark 1: Pointwise derivative vs. uniform contraction