DIGSP: Domain-Informed Genetic Superposition

• DIGSP is a symbolic regression framework tailored for engineering systems that decomposes inputs into domain-specific feature subsets and evolves independent GP populations.
• It independently evolves GP populations on separable physical mechanisms, using Elastic Net Regression for gene aggregation and ensemble fitness for synergy.
• DIGSP incorporates an adaptive hierarchical symbolic abstraction mechanism to filter, compress, and transfer significant expressions, yielding improved RMSE and model parsimony.

Domain-Informed Genetic Superposition Programming (DIGSP) is a symbolic regression framework tailored for engineering systems governed by separable physical mechanisms. It partitions the input space into domain-specific feature subsets and evolves independent genetic programming (GP) populations to model material-specific effects. Early evolution occurs in isolation, while ensemble fitness promotes inter-population cooperation. After stagnation across all populations, an adaptive hierarchical symbolic abstraction mechanism (AHSAM) performs ANOVA-based filtering, compresses statistically significant expressions into symbolic constructs, and injects them into all populations through a validation-guided pruning cycle. In a case study on steel fiber-reinforced concrete (SFRC) beams, DIGSP was benchmarked against a baseline multi-gene genetic programming model and consistently outperformed that baseline in training and test root mean squared error (RMSE), with Wilcoxon rank-sum results indicating statistical significance for train and test performance (Khorshidi et al., 20 Sep 2025).

1. Domain-informed decomposition and problem setting

DIGSP is designed for engineering problems where physical mechanisms are separable and superposition applies. Its central modeling assumption is that the target system can be decomposed into semantically coherent, physically meaningful domains rather than treated as a monolithic input space. In the SFRC beam case, the reported groupings include Fiber & Geometry, Concrete & Geometry, and Steel & Geometry. Examples given for these groups are fiber volume fraction $V_f$, fiber aspect ratio $l_f/d_f$, geometric variables $(a, d, a/d)$, compressive strength $f_c'$, aggregate size $d_f$, and reinforcement ratio $\rho$ (Khorshidi et al., 20 Sep 2025).

Each domain is assigned a separate, independently evolving GP population. In early evolution, each population operates solely on its feature subset, with no gene exchange across populations. The method is therefore explicitly structured around structural separability in the physical system. The reported significance of this design is that it respects mechanism-level decomposition from the outset, rather than attempting to recover it indirectly after unconstrained search.

This decomposition is not merely a feature-selection heuristic. DIGSP frames it as a mechanism-level partition of the symbolic search space. The stated motivation is that symbolic superposition should align with the underlying physical structure, and the framework is intended for systems in which that alignment is meaningful (Khorshidi et al., 20 Sep 2025).

2. Population architecture, gene aggregation, and ensemble fitness

Within each population, DIGSP evolves symbolic expressions using GP operators including initialization, selection, crossover, and mutation. The population structure is described as $P$ populations, each with individuals containing a fixed number of symbolic genes, with three given as an example. Each gene outputs a sub-expression, and these sub-expressions are linearly combined using Elastic Net Regression (ENR) for within-population aggregation, balancing sparsity and regularization (Khorshidi et al., 20 Sep 2025).

For sample $x_i$ in population $k$, the reported within-population predictor is

$$\hat{y}_i^{(k)} = \sum_{j=1}^{p} \beta_j^{(k)} g_j^{(k)}(x_i) + \beta_0^{(k)}.$$

The corresponding ENR optimization is

$l_f/d_f$0

DIGSP then introduces ensemble fitness. Outputs of top individuals from each population are linearly fused, again using ENR, to create a global prediction:

$l_f/d_f$1

The reported interpretation of this architecture is twofold. First, decomposability and interpretability are retained at the domain or mechanism level because each population remains anchored to a designated subset of variables. Second, synergistic improvements are enabled through ensemble learning, since the final prediction aggregates complementary domain-specific models rather than forcing a single symbolic structure to absorb all effects (Khorshidi et al., 20 Sep 2025).

3. Adaptive Hierarchical Symbolic Abstraction Mechanism

AHSAM is the mechanism by which DIGSP moves from isolated early evolution to explicit symbolic superposition. It is triggered when evolutionary progress stagnates across all populations for $l_f/d_f$2 generations. At that point, DIGSP initiates a two-phase process intended to enable knowledge exchange and hierarchical reuse of discovered symbolic mechanisms (Khorshidi et al., 20 Sep 2025).

In Phase I, DIGSP performs ANOVA-based symbolic filtering. The outputs of all individuals in all populations are collected, and each expression is evaluated using one-way ANOVA to determine whether it produces significantly distinct output distributions. The reported decomposition is

$l_f/d_f$3

with test statistic

$l_f/d_f$4

Individuals with $l_f/d_f$5 are retained.

In Phase II, each retained expression $l_f/d_f$6 is compressed and treated as a new high-level symbolic feature,

$l_f/d_f$7

These constructs are then injected into the terminal set of all populations. The injection is not unconditional. For each candidate $l_f/d_f$8, DIGSP performs a utility check comparing standalone RMSE against population mean RMSE. If $l_f/d_f$9 improves or does not worsen RMSE, it is retained; otherwise, destabilizing subtrees are pruned until utility is achieved. This cycle is validation-guided, meaning that pruning and acceptance are based on performance on a hold-out validation set.

A common simplification is to describe DIGSP as permitting direct inter-population gene exchange throughout evolution. The reported mechanism is narrower and more structured than that. Early evolution occurs in isolation, and cross-population reuse is introduced only after global stagnation, only for ANOVA-filtered expressions, and only through validation-guided pruning and injection. In that sense, AHSAM functions as an adaptive, statistically mediated form of symbolic transfer rather than unrestricted crossover (Khorshidi et al., 20 Sep 2025).

4. Termination criteria and model selection logic

DIGSP continues alternating ordinary evolution and possible AHSAM cycles until one of two stopping conditions is met: the maximum number of generations is reached, or no improvement in ensemble fitness is observed for a further $(a, d, a/d)$0 generations after AHSAM. The final model is then chosen according to the lowest ensemble RMSE on the validation set (Khorshidi et al., 20 Sep 2025).

This model-selection criterion is consistent with the framework’s broader use of validation-guided pruning in AHSAM. The same hold-out logic that governs whether symbolic constructs are accepted into terminal sets also governs selection of the final ensemble. The reported workflow therefore couples evolutionary search, inter-population cooperation, symbolic abstraction, and final selection through a single validation-centered decision rule.

The simplified algorithmic pseudocode supplied with the study presents the same logic operationally: partition variables into $(a, d, a/d)$1 domain-informed groups, initialize GP populations over those groups, evolve each population independently, aggregate top individuals into an ensemble prediction, trigger AHSAM after $(a, d, a/d)$2 generations of global stagnation, and terminate on convergence or the maximum generation count. The final model is selected by best validation RMSE (Khorshidi et al., 20 Sep 2025).

5. SFRC beam benchmark and statistical evaluation

The benchmark study used a dataset of 213 samples of SFRC beams, described as a well-established structural engineering benchmark. Variables included geometry, materials, fiber specifics, reinforcement, and related descriptors. Data partitioning was 65% training, 10% validation, and 25% testing, with random partitioning and 30 independent runs per configuration. The baseline for comparison was a multi-gene genetic programming model (BGP), described as standard GP with a single population and all features available for all genes (Khorshidi et al., 20 Sep 2025).

The primary evaluation metric was RMSE on training, validation, and test sets. The study also used boxplots, violin plots, and raincloud plots to visualize performance distributions. Statistical testing was performed with the Wilcoxon rank-sum test for paired differences in RMSE distributions. The reported example for training RMSE was $(a, d, a/d)$3, and the summary finding was Wilcoxon $(a, d, a/d)$4 for train and test RMSE (Khorshidi et al., 20 Sep 2025).

Across the 30 independent trials, DIGSP consistently outperformed BGP in training and test RMSE. The reported distributional pattern was lower mean and median RMSE on both training and test sets, fewer outliers, and tighter accuracy distributions. No significant difference was observed in validation RMSE, and the reported explanation was limited sample size. The study also compared parsimony through tree size, defined as number of nodes, and number of terms. DIGSP generated much smaller symbolic models, with a reported median tree size of 41 nodes versus 204 nodes for BGP (Khorshidi et al., 20 Sep 2025).

These results were presented as evidence that domain-informed structural decomposition and symbolic abstraction improve convergence and generalization. The evidence base reported in the study is therefore not limited to point estimates of predictive accuracy; it also includes repeated random splits, distributional visualization, formal nonparametric testing, and explicit model-complexity comparisons.

6. Interpretability, parsimony, and relation to adjacent symbolic-regression frameworks

DIGSP’s interpretability claim is tied directly to its decomposition strategy. Contribution analysis reportedly showed that DIGSP’s decomposed models map onto physical mechanisms—concrete, steel, and fiber—in line with engineering knowledge. The method’s stated interpretability therefore derives not only from symbolic form, but from the fact that symbolic submodels are assigned and evolved at the level of material-specific or mechanism-specific domains (Khorshidi et al., 20 Sep 2025).

The reported significance section attributes several effects to this structure. Domain-informed decomposition is said to yield faster, more stable convergence, suppress bloat, and align discovered models with physical or mechanistic understanding. AHSAM is described as a principled, adaptive “crossover” across domains guided by statistical evidence, increasing the utility and parsimony of transferred sub-expressions. DIGSP is also presented as generalizable beyond SFRC beams to decomposable physical systems in physics, materials, and biology (Khorshidi et al., 20 Sep 2025).

Within the broader symbolic-regression literature, DIGSP can be contrasted with the neural-network-directed genetic programmer faiGP. faiGP integrates a convolutional variational autoencoder, a Bayesian multilabel classifier, and GP guided by a grammar based on function algebra; its paper emphasizes neural-network-directed priors, set-based algebraic representation, universal approximation guarantees, empirical extraction of constants, and diversity regularization (Razavi et al., 2022). The comparison presented there places DIGSP among domain-informed symbolic-regression approaches, while distinguishing faiGP by those neural and algebraic components. This situates DIGSP as a framework centered on structural decomposition, population modularity, ENR-based aggregation, and statistically gated symbolic abstraction rather than neural prior learning (Razavi et al., 2022).

DIGSP is thus characterized in the literature as a modular, interpretable, and statistically guided symbolic regression framework for systems where symbolic superposition aligns with the underlying physical structure. Its defining elements are the combination of domain-informed partitioning, independently evolving GP populations, ENR-based ensemble fusion, and AHSAM-mediated symbolic reuse, together with benchmark evidence on SFRC beams showing improved training and test RMSE, tighter error distributions, and substantially smaller symbolic models relative to a baseline BGP configuration (Khorshidi et al., 20 Sep 2025).