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

This paper proposes a minimally structured Neural Ordinary Differential Equation architecture that enforces trajectory stability and parameterizes multistable nonlinear systems by learning a vector field with a contraction term and an implicit equilibrium map, enabling efficient data-driven discovery and gradient-based control.

The Gist

The Big Problem: The "Mystery Box" of Stable Systems

Imagine you are trying to figure out how a complex machine works, like a coffee maker or a thermostat. You can see the buttons you press (the control inputs) and the temperature of the coffee (the state).

In many engineering systems, things are "stable." If you set the thermostat to 70°F, the room eventually settles at 70°F and stays there. It doesn't spin out of control (chaos), and it doesn't oscillate wildly.

The Catch: Because these systems are so stable and settle down quickly, they don't give you much "action" to watch. It's like trying to guess the rules of a game by watching someone sit still on a bench. If you only watch them for a short time, you might think there are many different ways the game could be played, because you haven't seen the full picture. This makes it very hard for computers to learn the true rules of the system from just a little bit of data.

The Solution: A "Smart Map" with a Safety Net

The authors propose a new way for computers (specifically, a type of AI called a Neural ODE) to learn these systems. Instead of letting the AI guess the rules freely (which often leads to wild, unstable guesses), they force the AI to build a specific kind of "Smart Map."

They break the system's behavior into two parts, like a GPS and a Brake:

1. The GPS (The "Target" $g$): This part of the AI learns where the system wants to go. If you press a button, where does the system want to settle? This is the "equilibrium."
2. The Brake (The "Pull" $f$): This part of the AI learns how fast the system gets there. The authors force this part to always be negative (like a brake). This guarantees that no matter what happens, the system will always slow down and stop at the target. It prevents the AI from learning a model that explodes or goes crazy.

The Analogy: Imagine a marble rolling in a bowl.

• The shape of the bowl is the "GPS." It has hills and valleys. The marble wants to roll to the bottom of a valley.
• The friction is the "Brake." It ensures the marble doesn't bounce forever; it eventually stops.
• The authors' method teaches the AI to learn the shape of the bowl and the amount of friction, but it forces the friction to always be positive so the marble never flies off the table.

Why This is a Game-Changer: The "Hysteresis" Loop

Some systems are tricky. They have Multistability, meaning they can settle in different places depending on their history.

The Analogy: Think of a light switch that is a bit sticky.

• If you push it up, it clicks on.
• If you push it down, it clicks off.
• But if you push it just a little bit from the "on" position, it might not turn off until you push it way down.
• This is called Hysteresis. The path you took to get to the current state matters.

Most AI models struggle with this because they get confused by the "sticky" behavior. But because the authors' "Smart Map" explicitly separates the "target location" from the "speed of arrival," the AI can easily see the different valleys in the bowl. It can learn that "If you come from the left, the target is here; if you come from the right, the target is there."

The Superpower: Controlling the System

Once the AI has learned this "Smart Map," controlling the system becomes incredibly easy.

Usually, controlling a complex system is like trying to steer a ship in a storm by guessing which way to turn the wheel. You have to simulate the future, guess, and adjust.

With this new method, the AI has a direct map ($g$) that tells it exactly where the system will end up if you set a specific control.

• The Goal: "I want the room to be 72°F."
• The AI's Job: It looks at its map, finds the spot on the map that says "72°F," and simply calculates the button press needed to get there.
• The Result: It can steer the system through the "sticky" hysteresis loops effortlessly, even if the system is noisy or if you only have a tiny amount of data to learn from.

Real-World Examples They Tested

The team tested this on four different scenarios to prove it works:

1. Water Tanks: Two connected tanks where water flows in and out. They showed the AI could learn how the water levels settle and then control the pumps to hit specific water levels perfectly.
2. Symmetric Hysteresis: A mathematical model of a "tipping point" (like a climate system that flips between two states). The AI learned the "sticky" behavior and could push the system back and forth across the tipping point without breaking it.
3. Budworm Population: A model of insect outbreaks. These populations can suddenly explode or crash. The AI learned to predict these crashes and control the population to stay safe.
4. Genetic Toggle Switch: A synthetic biology system (like a light switch inside a cell). This is very complex with many variables. The AI successfully learned the complex "sticky" behavior and could flip the switch on and off reliably.

The Bottom Line

This paper introduces a "safety-first" way for AI to learn how stable systems work. By forcing the AI to learn a structure that guarantees the system will eventually settle down (stability), they solved the problem of "not enough data."

In short: They gave the AI a pair of training wheels that never come off. This allows the AI to learn complex, "sticky" systems quickly and then drive them exactly where we want them to go, even through the trickiest parts of the road.

Technical Summary

1. Problem Statement

The paper addresses the challenge of identifying and controlling asymptotically stable nonlinear dynamical systems that exhibit multistability (multiple equilibrium points) and hysteresis (history-dependent behavior).

• The Core Difficulty: Traditional system identification from data often fails for non-chaotic systems. Because stable dynamics converge quickly to equilibria, they provide limited excitation (data diversity), making the inverse problem of discovering the governing equations ill-posed and non-unique.
• The Control Challenge: Controlling such systems is difficult because standard feedback policies struggle to navigate bifurcations (tipping points) and hysteresis loops, where the system state depends on the history of control inputs.
• The Gap: Existing Neural Ordinary Differential Equation (NODE) frameworks often lack structural constraints to guarantee stability or fail to capture multiple attractors without extensive, long-duration data.

2. Methodology

The authors propose a minimally structured Neural ODE (NODE) architecture that enforces physical constraints directly into the model structure, rather than relying solely on loss functions.

A. Structured Dynamics Formulation

Instead of learning a generic vector field $F_\theta(x, u)$, the authors constrain the dynamics to the following form:
$$\frac{dx}{dt} = F(x, u) = f_\theta(x) \cdot (x - g_\theta(x, u))$$
Where:

• $f_\theta(x)$: A neural network constrained such that $f_\theta(x) < 0$ element-wise. This acts as a state-dependent decay rate, ensuring that trajectories contract toward the target defined by $g$.
• $g_\theta(x, u)$: A neural network representing the implicit equilibrium map. The fixed points of the system occur where $x = g_\theta(x, u)$.
• Significance: This structure guarantees trajectory stability (boundedness) across the entire state space. Crucially, because $g_\theta$ is learned independently of the decay rate, the model can naturally represent multiple attractors (multistability) and complex bifurcation structures like hysteresis loops.

B. Training Objectives

The model is trained using two potential objectives:

1. Trajectory Matching: Minimizing the difference between simulated trajectories and observed data (standard NODE approach).
2. Gradient Matching: Minimizing the difference between the learned vector field and the time-derivatives estimated from data (using finite differences). This is particularly useful for short time-horizon data where trajectory integration errors might accumulate.

C. Control Strategy

The learned equilibrium map $g_\theta$ enables efficient gradient-based feedback control:

• Goal: Drive the system to a desired state $x^*$.
• Mechanism: Since steady states satisfy $x^* = g_\theta(x^*, u)$, the control input $u$ is updated via gradient descent to minimize the distance between the current state and the target equilibrium:
  $$\frac{du}{dt} = -\eta \nabla_u \frac{1}{2} \| g_\theta^{\circ k}(x, u) - x^* \|^2$$
  Here, $g_\theta^{\circ k}$ denotes $k$-fold iteration of the map, acting as an inexpensive approximate forecaster.
• Constraints: The framework easily incorporates control constraints (e.g., physical bounds on inputs) using smooth Heaviside functions within the gradient update.

3. Key Contributions

1. Structured Stability: Introduction of a specific NODE architecture that mathematically guarantees asymptotic stability and bounded trajectories by construction, avoiding the non-convex loss landscapes typical of unstructured NODEs.
2. Multistability from Sparse Data: Demonstration that the model can identify systems with multiple basins of attraction and hysteresis using only short time-horizon data (transient dynamics), where traditional methods often fail due to lack of excitation.
3. Tractable Control: The decoupling of the decay rate ($f$) and the equilibrium map ($g$) allows for direct optimization of control inputs to navigate complex bifurcation diagrams and tipping points without solving the full ODE forward in time for every control step.
4. Interpretability: The learned $g_\theta$ directly reveals the system's bifurcation diagram and tipping points, offering physical interpretability often lost in black-box deep learning models.

4. Numerical Results

The method was validated on four distinct nonlinear benchmarks:

• Mixing Tanks (Hydroelectric Load Balancing):
  • Successfully learned dynamics from short trajectories ($t=200$ vs. full evolution $t=1000$).
  • Achieved low normalized RMSE ($\approx 1.5\%$) in stochastic feedback control, successfully steering the system to randomized targets.
• Symmetric Hysteresis (Cubic Bifurcation):
  • Learned the bistable regime and tipping points of the system $\dot{x} = \lambda + x - x^3$.
  • The model correctly identified the zero-level sets of the equilibrium map, reproducing the bifurcation diagram.
  • Control policies successfully navigated the system across tipping points to reach unstable equilibria.
• Budworm Population (Asymmetric Hysteresis):
  • Tackled a classic insect outbreak model with asymmetric steady-state branches.
  • The model avoided the common pitfall of learning only the dominant branch, capturing the full hysteretic loop despite data scarcity.
• Genetic Toggle Switch (High-Dimensional Bistability):
  • Modeled a 2D gene-regulatory network with 4 control parameters.
  • Demonstrated the ability to learn coupled hysteretic bifurcations and steer the system to targets using constrained control policies (enforcing cooperativity coefficients $>1$).
  • Achieved high accuracy (99% of trials within 5% error) despite complex coupled dynamics.

5. Significance and Impact

• Bridging Data Scarcity and Stability: This work provides a solution for engineering and biological systems where collecting long-term trajectory data is impossible or impractical. By enforcing stability structurally, the model learns effectively from transient data.
• Control of Complex Systems: It offers a new paradigm for controlling systems with hysteresis and tipping points (e.g., climate models, chemical reactors, biological switches), where traditional linear control or standard deep learning often fails to generalize across bifurcations.
• Scientific Interpretability: The approach moves beyond "black box" prediction to "white box" discovery, explicitly recovering the underlying bifurcation structure and equilibrium manifolds, which is critical for scientific understanding and safety-critical applications.

In summary, the paper presents a robust framework that combines structural priors with deep learning to solve the difficult problems of identifying and controlling multistable, hysteretic systems from limited data, with direct applications in engineering, biology, and physics.