Guaranteed Conformance of Neurosymbolic Models to Natural Constraints

Abstract: Deep neural networks have emerged as the workhorse for a large section of robotics and control applications, especially as models for dynamical systems. Such data-driven models are in turn used for designing and verifying autonomous systems. They are particularly useful in modeling medical systems where data can be leveraged to individualize treatment. In safety-critical applications, it is important that the data-driven model is conformant to established knowledge from the natural sciences. Such knowledge is often available or can often be distilled into a (possibly black-box) model. For instance, an F1 racing car should conform to Newton's laws (which are encoded within a unicycle model). In this light, we consider the following problem - given a model $M$ and a state transition dataset, we wish to best approximate the system model while being a bounded distance away from $M$. We propose a method to guarantee this conformance. Our first step is to distill the dataset into a few representative samples called memories, using the idea of a growing neural gas. Next, using these memories we partition the state space into disjoint subsets and compute bounds that should be respected by the neural network in each subset. This serves as a symbolic wrapper for guaranteed conformance. We argue theoretically that this only leads to a bounded increase in approximation error; which can be controlled by increasing the number of memories. We experimentally show that on three case studies (Car Model, Drones, and Artificial Pancreas), our constrained neurosymbolic models conform to specified models (each encoding various constraints) with order-of-magnitude improvements compared to the augmented Lagrangian and vanilla training methods. Our code can be found at: https://github.com/kaustubhsridhar/Constrained_Models

• The paper introduces a memory-based partitioning method that guarantees deep neural networks adhere to natural constraints derived from symbolic models.
• It reduces approximation errors by bounding them through increased memory partitions, ensuring reliable predictions in complex systems.
• Experimental validations on vehicles, medical simulators, and drones show significant reductions in constraint violations compared to traditional methods.

Guaranteed Conformance of Neurosymbolic Models to Natural Constraints

The paper presents a novel approach to integrating deep neural networks (DNNs) with symbolic models to ensure conformance with natural constraints in modeling dynamical systems. It addresses the challenges of deploying neural networks in safety-critical applications, where the models must comply with established scientific principles. This method is of particular relevance in fields like robotics and medical systems, where ensuring adherence to theoretical models is crucial for reliable and predictable behavior.

The authors introduce a framework that guarantees DNNs respect constraints derived from symbolic models, which could be black-box representations like physical laws or physiological properties. At the core of this framework is using a memory-based partitioning method that utilizes neural gas algorithms to distill input data into representative subsets. These subsets enable the computation of bounds for the neural network’s predictions, ensuring that each subset’s predictions comply with the defined constraints.

The methodological innovation is particularly significant in ensuring smaller approximation errors by controlling them with the number of memory partitions. The authors derive theoretical guarantees, showing that the increase in approximation error due to these constraints is bounded and manageable by introducing more partitions. This is a critical assurance in practical applications, as it suggests that the approach scales favorably with model complexity and computational resources.

The experimental setup is robust, with case studies spanning a variety of complex dynamical systems: a vehicle model in CARLA, a glucose-insulin dynamics model for artificial pancreases, and drones in the Pybullet framework. Across these studies, the proposed constrained neurosymbolic models demonstrate significant improvements in adhering to constraints compared to traditional methods like augmented Lagrangian. These models conform to natural constraints, yielding manifold reductions in constraint violations across several metrics.

An intriguing aspect of this study is the emphasis on using black-box models for constraints, which mirrors real-world applications where precise models might not be explicit. The focus on memory-based partitions highlights an elegant balance between computational feasibility and theoretical rigor, showing potential beyond current applications.

In terms of implications, this work paves the way for more reliable integration of neural networks in systems that require strict adherence to foundational scientific principles. Its potential extends to creating neurosymbolic policies that could lead to safer and more efficient autonomous systems in complex environments.

Future research could explore the scalability of the method further and integrate these constrained models into broader control architectures, examining how they might interact with feedback mechanisms and adaptive strategies in real-time environments. The theoretical robustness and practical efficiency suggest a promising path forward in neurosymbolic systems’ convergence with established physical principles.