Source-linked AI summary
Learning Equations for Extrapolation and Control
Subham S. Sahoo, Christoph H. Lampert, Georg Martius
TL;DR
The paper addresses how to learn concise, interpretable equations from noisy data and extrapolate them beyond observed domains. It introduces EQL÷ with differentiable division units and improved model selection, then demonstrates equation learning and robot control, including swing-up from two random rollouts.
Problem
Black-box regression generally evaluates prediction on unseen data from the same distribution, while identifying interpretable equations requires recovering functional relationships that support generalization to unseen parameter-space regions.
Method
EQL÷ extends equation-learning networks with differentiable division units, curriculum regularization for stable training, and model selection based on interpolation validation and sparsity.
Results
The method learns concise functional relations from noisy data, extrapolates to unseen domains, and identifies cart-pendulum forward dynamics from 2 random rollouts sufficient for robust swing-up control.
Takeaways & Limitations
EQL÷ can provide inspectable equations and enable control of an actuated cart-pendulum system even when the swing-up pattern was absent from training.
Takeaways & Limitations
The regression formulation assumes data arise from an unknown analytical function with additive zero-mean noise, and optimization can get stuck in local minima or select an incorrect formula.
Abstract
from arXiv · showhide
We present an approach to identify concise equations from data using a shallow neural network approach. In contrast to ordinary black-box regression, this approach allows understanding functional relations and generalizing them from observed data to unseen parts of the parameter space. We show how to extend the class of learnable equations for a recently proposed equation learning network to include divisions, and we improve the learning and model selection strategy to be useful for challenging real-world data. For systems governed by analytical expressions, our method can in many cases identify the true underlying equation and extrapolate to unseen domains. We demonstrate its effectiveness by experiments on a cart-pendulum system, where only 2 random rollouts are required to learn the forward dynamics and successfully achieve the swing-up task.
1. Introduction
The paper develops an interpretable equation-learning approach that extends prior work to divisions, improves model selection, and demonstrates robot control from very little data.
- Motivation: Unlike black-box regression, the approach seeks interpretable functional relationships that can generalize beyond the observed data distribution.It targets models that support understanding of systems and hypotheses about unseen situations.
- Motivation: EQL previously represented plausible functions with sparsity regularization but could not represent divisions, limiting its applicability to physical systems.
- Contributions: EQL÷ introduces a network architecture that handles divisions and techniques for stable training.
- Contributions: The paper improves model and instance selection to identify the appropriate network or equation more effectively.
- Contributions: The method learns forward dynamics equations for a robotic system from very few random trials and uses them to control the system.
2. Identifying equation with a network
EQL÷ extends equation-learning networks with division units and training safeguards, while combining sparsity with validation error to select equations that can extrapolate beyond observed data.
- Problem: The method models noisy observations from an unknown analytical function while targeting both training accuracy and performance on future data from unvisited feature-space regions.This supports applications such as learning robot dynamics in safe regions and predicting movements elsewhere.
- Architecture: EQL÷ uses algebraic-function units in a feed-forward network and adds division operations in the output layer to avoid divergences during optimization.The architecture extends EQL with division while restricting divisions to the output layer and using the positive branch of 1/b.
- Training: Regularized division sets outputs to zero below a denominator threshold, penalizes forbidden denominators, and gradually reduces the threshold through a training curriculum.The zero output produces zero gradient for forbidden denominator values, while penalty epochs sample unlabeled inputs from the expected test and extrapolation range.
- Training: Training combines L2 loss, L1 sparsity regularization, and denominator penalties, with staged regularization that first frees weights, then encourages sparsity, and finally fixes the L0 structure.The staged scheme is intended to produce sparse connections corresponding to the few terms in physical-system formulas.
- Model selection: The Vint-S method selects models by combining normalized interpolation-validation error and sparsity, weighted equally with α = 0.5 and β = 0.5.This replaces rank-based selection because ranking can ignore numeric validation-error differences and distinguish nearly equivalent instances.
- Model selection: Around 40 unlabeled points from the extrapolation domain were sufficient to identify the correct instance from more than 100 candidate equations.The selection relies on the large extrapolation-error difference between correct and incorrect formulas.
3. Relation to prior work
The paper frames equation learning as discovering interpretable input-output relationships rather than merely fitting black-box predictors. It combines learned base-function compositions with a gradient-based alternative to symbolic search and introduces division units as a new neural-network capability.
- Equation learning seeks concise, interpretable functions that reveal underlying relationships and support reasoning about unseen situations.
- Unlike system identification, the method learns both base-function parameters and their composition.
- Symbolic regression can discover physical laws but faces prohibitive exponential search complexity for larger expressions and high-dimensional problems.
- The approach differs from causal learning by targeting exact functional dependencies rather than only causes, effects, or graphical factorization.
- Domain-adaptation methods are not directly applicable because the training and extrapolation domains are disjoint, despite sharing a labeling function.
- The paper combines previously introduced components into a new architecture and reports the first neural network use of division units.
4. Experimental evaluation
Experiments test equation learning on division, complex, randomly generated, and physical-system tasks under restricted training domains and larger extrapolation regions. EQL÷ generally achieves the strongest extrapolation, including reliable division recovery and successful learning of cart-pendulum dynamics, though one run performs poorly.
- Experiments use restricted noisy training domains, interpolation and extrapolation test sets, and comparisons against EQL, MLP, SVR, and Eureqa.The evaluation covers formula-learning and physical-system settings.
- 4.1. Learning formulas with divisions: Only EQL÷ extrapolates the division expression across all 10 independent runs, while original EQL lacks division units and Eureqa misses the right equation in 4/5 cases.All methods interpolate at the noise level of 0.01.
- 4.2. Learning complex formulas.: 15 times larger than the training domain, the complex-formula setting challenges models to extrapolate from a 4-dimensional hypercube.The correct formulas require between one and three hidden layers, with layer count selected automatically.
- 4.2. Learning complex formulas.: EQL÷ and Eureqa achieve good extrapolation in all tested formula cases, while original EQL finds the right answer only for F-2.With Vint&ex model selection, EQL÷ and Eureqa find F-3 every time and F-4 in about half the cases.
- 4.4. Cart-pendulum system: EQL÷ predicts the cart-pendulum function well beyond the training region, whereas other methods differ considerably from the true values further away.EQL÷ performs poorly in 1 of 10 independent runs; the paper reports this is less likely with a finer scan of λ values.
5. Control using learned dynamics
The paper learns cart-pole forward dynamics from random interaction data and uses the learned models for model predictive control. EQL÷ achieves swing-up with very few rollouts, whereas the MLP baseline does not stabilize the pole upright.
- Learning and control: The learner models cart-pole forward dynamics from state-action interactions and subsequently uses the model for swing-up control.The state includes cart position and velocity, pole angle and angular velocity; the learned model predicts state changes.
- Data collection: K − 1 random rollouts collect training data, followed by one additional rollout for model selection.Training rollouts use random actions drawn from N(0, 0.15), while the selection rollout uses N(0, 0.25).
- Control method: Model predictive control simulates 1000 random-action lookaheads per timestep, applies the first action from the lowest-cost trajectory, and runs at a 100 Hz update rate.The desired state is an upright pole with small velocities and the cart at the center.
- Results: K = 2 rollouts suffice for swing-up in most cases, while K = 3 supports evaluation of robustness to sensor and action noise.The K = 2 setting uses one training and one validation rollout.
- Results: The EQL÷ controller robustly performs swing-up, whereas the MLP only swings the pole and does not stabilize it in the vertical position.The comparison is reported for learned forward models used in the swing-up task.
6. Conclusions
The paper introduces EQL÷ to learn concise equations that include divisions and improves model selection for identifying functional relations. Experiments show extrapolation to unseen domains and robot control from only two random rollouts.
- Contributions: EQL÷ extends equation-learning networks with division while remaining fully differentiable and trainable by back-propagation.A regularized division is smoothly transformed toward true division during curriculum training.
- Contributions: A new model-selection technique identifies the actual functional relation more reliably than purely sparsity-based selection.
- Empirical findings: EQL÷ learns functional relations with and without divisions from noisy data in confined domains and extrapolates them to unseen parts of the data space.The learned forms are concise and inspectable.
- Robot control: EQL÷ identifies an actuated cart-pendulum forward model from 2 random rollouts with limited excitation and noisy observations, then performs swing-up despite no such pattern during training.