🤖 machine learning

Learning piecewise-smooth dynamical systems

This paper presents a modular framework for discovering piecewise-smooth dynamical systems from trajectory data by combining statistical analysis of switching hyperplane identifiability with geometry-constrained neural networks to learn governing equations and characterize discontinuous behaviors like sliding motion.

Original authors: Davide Murari, Erik Jansson, Chris Budd OBE, Carola-Bibiane Schönlieb

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Davide Murari, Erik Jansson, Chris Budd OBE, Carola-Bibiane Schönlieb

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the physical world, many systems do not change their behavior gradually. A heavy box pushed across a floor does not slide smoothly from the start; it sits still, resisting every push, until the force becomes great enough to break the static grip, and then it suddenly moves. A stiff door hinge does not turn with a fluid motion but catches and releases in tiny, jerky steps. These are examples of systems that switch between distinct modes of behavior. In mathematics, such systems are described by rules that are smooth and predictable within each mode, but which jump abruptly when the system crosses a specific boundary. These boundaries are not just lines on a graph; they are the physical thresholds where friction gives way to motion, or where a climate system shifts from one stable state to another. Understanding these systems is crucial for engineers designing robots, for climate scientists modeling ice ages, and for anyone trying to predict how a complex machine will behave when it hits a limit. However, when scientists try to learn the rules of such systems from observed data, they face a unique challenge: if they get the location of the boundary wrong, the entire prediction fails, no matter how well they understand the motion on either side.

A team of researchers at the University of Cambridge and the University of Bath has developed a new method to solve this problem. Their work focuses on discovering the hidden rules of these "piecewise-smooth" systems directly from trajectory data, which are simply records of how a system moves over time. Instead of trying to guess the entire system at once, the researchers split the task into two distinct stages. First, they look for the boundaries themselves. They analyze the recorded paths to find sudden jumps in speed or direction, which act as signals that the system has crossed a threshold. Using a robust statistical technique, they filter out the noise and the false alarms caused by rough data to pinpoint the exact location and angle of these invisible dividing lines. Once the geometry of the boundaries is established, the second stage begins. The researchers then teach a computer model to learn the specific rules of motion for each region separated by those boundaries. Crucially, they designed the computer model so that it is physically impossible for it to invent new, fake boundaries. It is forced to respect the map created in the first stage, learning only how the system moves within the allowed zones.

The researchers tested this two-stage approach on several real-world scenarios, including a mechanical oscillator that mimics the stick-slip motion of a block sliding on a rough surface, and a complex climate model that describes the cycle of ice ages over thousands of years. In the climate model, the system switches between periods of slow carbon dioxide absorption and sudden release, a process that drives the growth and retreat of ice sheets. In every test, the method proved highly effective. When the researchers knew the true boundaries in advance, their model learned the motion with extreme precision, reproducing the system's behavior with errors far smaller than those of standard smooth models. Even more impressively, when they had to discover the boundaries from scratch using noisy data, the method still succeeded. It correctly identified the location of the switching planes and learned the dynamics well enough to predict the system's future path for hundreds of time steps. The results showed that by separating the task of finding the map from the task of learning the journey, the researchers could build models that were not only more accurate but also far more efficient, using significantly fewer computational resources than traditional methods.

The study also revealed the limits of what can be learned. The researchers found that the method works best when the system actually crosses the boundaries with enough force to create a clear signal. If the system merely grazes the boundary or moves very slowly across it, the signal becomes too weak to distinguish from the background noise, and the method struggles to find the line. However, when the crossing is clear, the approach is robust. The team demonstrated that their new architecture could approximate the complex, jumping behavior of these systems with arbitrary accuracy, provided the boundaries were known or correctly identified. This represents a significant step forward in the field of scientific machine learning, where the goal is to build models that respect the known physical structures of the world rather than just memorizing patterns in data. By explicitly teaching the computer to look for and respect the places where the rules change, the researchers have created a tool that can uncover the hidden logic of systems that jump, slide, and switch, offering a clearer window into the mechanics of everything from dry friction to the shifting climate of the Earth.

Technical Summary: Learning Piecewise-Smooth Dynamical Systems

Problem Statement
The paper addresses the challenge of discovering dynamical systems directly from trajectory data when the underlying dynamics are discontinuous. While data-driven methods have advanced significantly for smooth systems, they often struggle with piecewise-smooth systems (Filippov systems) common in applications like mechanical friction, sliding-mode control, and climate dynamics (e.g., glacial cycles). In these systems, the vector field is smooth within distinct polyhedral regions but jumps across switching hyperplanes. The core difficulty lies not only in learning the smooth dynamics within each regime but also in accurately detecting the geometry of the switching hyperplanes and characterizing behaviors such as sliding motion. A misplaced switching hyperplane invalidates the model regardless of the accuracy of the local dynamics.

Methodology
The authors propose a modular, two-stage framework that separates "geometry learning" from "dynamics learning."

Geometry Learning (Switching Hyperplane Recovery):
- The first stage estimates the arrangement of affine switching hyperplanes from trajectory data.
- The method relies on detecting jumps in estimated velocities. It employs a Savitzky–Golay differentiator to compute velocity estimates from noisy observations, then calculates velocity jumps ( $\Delta v$ ) between consecutive samples.
- Candidate points near discontinuities are identified by retaining intervals with the largest jump magnitudes and their neighbors.
- To robustly fit the hyperplane amidst outliers (caused by noise, curvature, or estimation artifacts), the authors use Random Sample Consensus (RANSAC) to find a coherent subset of candidates, followed by Total Least Squares (TLS) refinement.
- This stage can be skipped if the switching geometry is known a priori.
Dynamics Learning (Geometry-Preserving Neural Networks):
- Once a partition (either recovered or prescribed) is established, the second stage learns the vector field within each region.
- The authors introduce a novel neural network architecture designed to be "geometry-preserving." The network is constructed such that its discontinuities are strictly confined to the prescribed hyperplane arrangement.
- The architecture utilizes a specific gating mechanism based on the Heaviside step function applied to the signed distances from the hyperplanes ( $z_P(x)$ ). This allows the network to switch between different smooth sub-networks ( $\Phi$ and $\Psi$ ) without introducing spurious discontinuities elsewhere.
- Training minimizes a composite loss function including trajectory matching (rollout error), vector field supervision, and directional regularizers (normal and tangential penalties) to ensure consistency with Filippov sliding dynamics.

Key Contributions

Two-Stage Framework: A "geometry-first" approach that recovers switching hyperplane arrangements before fitting vector fields, allowing for independent inductive biases for geometry and dynamics.
Detectability Analysis: A theoretical analysis quantifying the conditions under which switching hyperplanes are identifiable from noisy data. It establishes a relationship between jump magnitude, observation noise, derivative estimation error, and the normal crossing velocity, showing that a sufficient "jump-to-noise" ratio is required for reliable recovery.
Geometry-Preserving Architecture: A novel neural network class that approximates piecewise-smooth vector fields to arbitrary accuracy on compact sets while guaranteeing that discontinuities occur only on the prescribed switching surfaces. The authors prove that this class can represent piecewise-affine fields exactly and approximate general piecewise-smooth fields without creating new jumps.
Empirical Validation: Extensive experiments on dry-friction oscillators and the PP04 climate model demonstrate that the proposed method achieves lower rollout and vector-field errors than smooth Neural ODE baselines, often using an order of magnitude fewer trainable parameters.

Results

Geometry Recovery: The RANSAC-based recovery successfully identifies switching hyperplanes in the presence of moderate noise. However, performance degrades when the jump magnitude is small relative to noise or when trajectories graze the switching surface (low normal velocity).
Dynamics Learning:
- Known Geometry: When the switching geometry is provided, the proposed architecture perfectly confines discontinuities to the correct surfaces, whereas unstructured jump networks produce spurious jumps.
- Recovered Geometry: When geometry is learned from data, the resulting models produce accurate long-horizon rollouts, provided the derivative-jump signal is strong enough.
- Structural Priors: Enforcing physical constraints (e.g., second-order structure $\dot{q}=p$ in friction models) significantly improves accuracy and generalization.
- Comparison: The discontinuous models outperform smooth Neural ODE baselines across all metrics (rollout error, final state error, vector field error) on both dry-friction and climate benchmarks, despite having significantly fewer parameters.

Significance and Claims
The paper claims that explicitly learning and preserving the switching geometry offers a practical and interpretable route to modeling discontinuous dynamics. By decoupling geometry recovery from dynamics learning, the method allows for the incorporation of domain knowledge at specific stages (e.g., known physics in the dynamics or known switching planes). The theoretical guarantees on the approximation properties of the proposed network class ensure that the model does not introduce artificial discontinuities, a common failure mode in standard neural approximations of non-smooth systems. The work is presented as a step toward robust scientific machine learning for systems where smoothness assumptions are invalid, with future work planned to address higher dimensions, interacting switches, and non-affine surfaces.

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