Additive Atomic Forests for Symbolic Function and Antiderivative Discovery
Authors: Reda Belaiche
Summary
The authors built a framework for discovering symbolic mathematical functions and their antiderivatives (integrals) simultaneously from data. The key insight is that the product and chain rules from calculus naturally generate function-derivative pairs that form a self-expanding library. They use two "primitives" (exponential-log and sine-cosine combinations) as seeds, then build "additive atomic forests"—sums of expression trees whose derivatives are fitted to data. Because derivatives are determined by construction, you automatically get both the function and its derivative without needing symbolic integration.
Main takeaways:
- Simultaneously discovers a function and its antiderivative from data using derivative algebra
- Self-expanding library grows via product rule and chain rule applied to elementary functions
- Two primitives (EML for e^u - ln v, SOL for sin u - cos v) seed the library efficiently
- "Additive atomic forests" are sums of expression trees fitted to data
- No symbolic integration step needed; derivatives determined by construction
- On 17 classification benchmarks, matches or exceeds XGBoost on 13 datasets while producing interpretable formulas
Relevance
Not related to my persona or language model work—this is about symbolic regression and mathematical function discovery. Included because it's a general-purpose interpretable ML technique.
Threat model
Potential threat/caveat for clean result "Fine-tuning one persona on a two-marker chunk and another on the start marker plants the end marker at every donor answer's end, not chained to the start (LOW confidence)": this item discusses benchmark.
Abstract
arXiv:2605.08130v1 Announce Type: new Abstract: We present a framework for the simultaneous symbolic recovery of a function and its antiderivative from data. The framework rests on three ideas. First, a derivative algebra: the observation that the product rule $\frac{d}{dx}[f \cdot g] = f'g + fg'$ and the chain rule, applied to a seed set of elementary functions, generate a self-expanding system of function-derivative pairs -- a living library that grows each time a new function is discovered. Second, two complementary primitives -- EML$,(e^u - \ln v)$, which is theoretically complete for all elementary functions, and SOL$,(\sin u - \cos v)$, introduced here, which makes trigonometric atoms available at depth1 instead of depth$\sim$8 -- that seed the library with core atoms cheaply. Third, additive atomic forests: finite sums of primitive trees, optionally composed via multiplicative nodes, whose derivatives are fitted to data by continuous optimisation or by exhaustive search over the library. Because differentiation of each atom is determined by construction, the forest simultaneously encodes a symbolic expression $F$ and its derivative $F'$; no symbolic integration step is required. The library is not a fixed object: it self-constructs from a small seed set by recursive application of the product rule, chain rule, and the two primitives, and it can grow as newly discovered functions are folded back in. The larger the library, the richer the expressible class of candidate functions. We give conditional completeness, additive-depth, and analytic simultaneous-recovery results for the framework. Empirically, in our reported runs on 17 classification benchmarks, sparse atom combinations match or exceed XGBoost on 13 datasets while producing interpretable formulas.