Neural networks are universal approximators — they can represent any continuous function. But they can't tell you what function they've learned. Symbolic regression can tell you the function but can't scale to the dimensionality and complexity that neural networks handle. The two approaches occupy opposite ends of a trade-off between expressiveness and interpretability.
The paper on Symbolic-KAN (arXiv: 2603.23854) builds a bridge using Kolmogorov-Arnold Networks — architectures where the learnable parameters are univariate functions on the edges, not multivariate weights at the nodes. The representation theorem guarantees that any multivariate function can be decomposed this way, but the practical challenge is making the learned univariate functions interpretable.
Symbolic-KAN adds gating mechanisms and symbolic regularization that push the continuous edge functions toward discrete symbolic primitives — sines, exponentials, polynomials. During training, the network is continuous and differentiable. After training, the converged edge functions can be read off as closed-form expressions, yielding an interpretable symbolic model.
The framework recovers governing equations in regression tasks, dynamical systems, and physics-informed learning of PDEs. The symbolic structure isn't imposed top-down — it emerges from the regularized training process.
The through-claim: interpretability and accuracy are not in fundamental tension — they're in optimization tension. A neural network trained without structural bias finds accurate but opaque solutions. The same network trained with symbolic regularization finds solutions that are both accurate and readable, because the true governing equations are themselves symbolic. The regularization doesn't reduce the solution space to something interpretable; it biases the optimizer toward the part of the solution space where the ground truth already lives.
2603.23854. Machine learning / symbolic regression / Kolmogorov-Arnold networks / interpretability / scientific discovery.