Multi-view Symbolic Regression (MvSR)

• Multi-view Symbolic Regression is a framework that extends traditional symbolic regression by fitting a common symbolic structure with independent, view-specific parameters.
• It employs aggregation functions like maximum or sum to ensure uniform model performance across diverse datasets and experimental conditions.
• Applications in astrophysics, finance, and chemistry demonstrate MvSR’s ability to improve accuracy and robustness over single-view approaches.

Multi-view Symbolic Regression (MvSR) is a generalization of symbolic regression for settings in which multiple datasets describe the same underlying law under different conditions, parameter settings, regimes, or observational contexts. Instead of returning a single formula specialized to one dataset, MvSR searches for a shared symbolic structure $f$ whose numerical parameters can be optimized independently for each view, thereby producing a parametric family of functions $f(x;\theta)$ rather than a one-off fit (Russeil et al., 2024). In the literature, this core multi-dataset formulation coexists with broader usages of “view,” including contiguous regime partitions of one feature space, multimodal pairings of numeric and symbolic representations, and factorized conditional decompositions; taken together, these usages place MvSR at the intersection of interpretable model discovery, multi-task fitting, and structured representation learning (Deshpande et al., 27 Feb 2026).

1. Definition and scope of the “multi-view” setting

In the original formulation, a view is one dataset corresponding to one experiment, one condition, or one object. If the $i$-th view is

$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$

MvSR assumes that all views share the same symbolic structure but admit different parameter vectors: $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$ The target is therefore a single symbolic expression $f(x;\theta)$ that acts as a parametric family, with view-specific parameters $\theta^{(i)}$ fitted independently for each dataset (Russeil et al., 2024).

This differs from classical symbolic regression, which solves a single-view problem of the form

$\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$

under the implicit assumption that all observations arise from one experimental setup. MvSR replaces that assumption with a two-level objective: $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$ where $\mathrm{agg}$ can be $f(x;\theta)$0, $f(x;\theta)$1, $f(x;\theta)$2, $f(x;\theta)$3, or harmonic mean; the original experiments use $f(x;\theta)$4, which minimizes the worst per-dataset fit and discourages solutions that explain only a subset of views (Russeil et al., 2024).

The meaning of “view” is not uniform across the SR literature. In the SDSS DR17 star/galaxy/quasar study, for example, the views are three contiguous, non-overlapping redshift segments of the same one-dimensional feature space $f(x;\theta)$5: a low-$f(x;\theta)$6 stellar-dominated regime, an intermediate-$f(x;\theta)$7 galaxy-dominated regime, and a high-$f(x;\theta)$8 quasar-dominated regime (Deshpande et al., 27 Feb 2026). In multimodal work such as MMSR and ViSymRe, “views” denote heterogeneous representations of the same underlying function, such as numeric support points, symbolic expressions, and visual encodings (Li et al., 2024). This suggests that MvSR has evolved from a narrow multi-dataset formulation into a broader family of shared-structure SR methods.

2. Optimization architecture and implementation variants

The defining algorithmic move in MvSR is the separation of symbolic structure from view-specific parameter estimation. A candidate expression tree or DAG is proposed by a search procedure, then its numerical constants are optimized independently in each view, and finally the per-view losses are aggregated into one fitness value. In the original Operon-based implementation, expressions are represented as GP trees; for each dataset, parameters are fitted by Levenberg–Marquardt nonlinear least squares, and the aggregated loss is used for selection, crossover, and mutation (Russeil et al., 2024).

The proof-of-concept MvSR implementation was explicitly described as satisfying three core requirements: it receives multiple datasets as input, optimizes parameters independently per dataset, and uses an aggregation function to compute an overall loss. The same source also identifies additional design goals—control of the maximum number of parameters, repeated parameters, and parameter-count penalties—but notes that these were not yet enforced in the initial implementation (Russeil et al., 2024).

A later comparative study in physical sciences makes clear that current MvSR-capable systems differ substantially in representation, parameterization, and aggregation (Russeil et al., 1 Sep 2025).

System	Search / representation	Multi-view handling
Operon	GP expression trees	View-specific parameters; worst-fitness $f(x;\theta)$9 aggregation
PySR	Multi-population evolutionary GP	Parametrized Template Expressions; view-specific parameters plus shared constants; sum aggregation
PhySO	RNN-generated symbolic expressions	Class SR with realization-specific parameters and class-parameters; sum aggregation
eggp	GP over an e-graph	View-specific parameters with parameter reuse; worst-fitness $i$0 aggregation

These implementation choices matter. Operon does not support limiting the maximum number of parameters, and each parameter cannot appear more than once per solution. PySR allows the same parameter to appear multiple times within a single equation and supports shared constants across views, but in the comparison it was allowed an unrestricted number of constants because that framework did not provide explicit control over this quantity. PhySO distinguishes realization-specific parameters from class-parameters optimized once across all datasets. eggp combines multi-view fitting with an e-graph representation intended to compactly store equivalent expressions (Russeil et al., 1 Sep 2025).

From a methodological standpoint, the aggregation function is the clearest fault line. Worst-fitness aggregation emphasizes uniform adequacy across views; sum or mean aggregation is more tolerant of heterogeneous view difficulty. The physical-sciences comparison recommends treating aggregation as an explicit hyperparameter rather than a fixed design choice (Russeil et al., 1 Sep 2025).

3. Empirical behavior in synthetic and real-world multi-dataset problems

The original MvSR paper validates the method on synthetic families where the true generating law is known. Three ground-truth expressions were used: a cubic polynomial, a Friedman-like function, and a more complex Friedman-based expression. For each function, four views were generated by varying parameters or restricting domain coverage; experiments varied noise level, maximum tree size, and random seed (Russeil et al., 2024). The key qualitative finding is consistent across these benchmarks: MvSR recovers the correct expression more frequently than single-view SR, especially when individual views are partial, noisy, or parameter-sparse.

A particularly revealing synthetic setup uses a cubic polynomial where different views set different coefficients to zero. In that case, single-view SR often prefers lower-degree polynomials that fit an individual dataset well but do not recover the full shared structure. MvSR, by contrast, consistently identifies the complete cubic in noiseless settings and remains more robust when noise and allowed tree size increase (Russeil et al., 2024). The same study reports that MvSR is significantly closer to the true number of parameters than single-view baselines, indicating better control of under- and over-parameterization.

On real data, MvSR was used to recover or extend compact parametric laws across chemistry, finance, and astrophysics. In absorption spectroscopy, with each molecule treated as a view, MvSR discovered

$i$1

which reduces to Beer’s law when $i$2 and saturates for $i$3; the reported fits for four molecules yield $i$4 (Russeil et al., 2024). In finance, using daily return distributions from multiple assets as views, MvSR rediscovered Gaussian-like, Laplace, and Cauchy forms and also proposed new Laplace-like variants, including the “Power-Laplace”

$i$5

which achieved the best median MSE, $i$6, among the tested return-distribution models (Russeil et al., 2024). In supernova light curves, MvSR recovered a Bazin-like function,

$i$7

and also identified a three-parameter exponential alternative,

$i$8

with mean $i$9 across six light curves (Russeil et al., 2024).

These results support the original motivation for MvSR: a shared symbolic law can be more identifiable when multiple partial or condition-specific views are fitted simultaneously than when each dataset is treated as an isolated regression problem.

4. Comparative performance in physical-science workflows

A broader implementation study evaluates Operon, PySR, PhySO, and eggp on five real-world multi-view datasets: supernovae light curves, galaxy rotation curves, Michaelis–Menten enzyme kinetics, the Nikuradse friction dataset, and network degree distributions (Russeil et al., 1 Sep 2025). Each dataset is explicitly cast as a multi-view problem, with views corresponding to individual light curves, rotation curves, concentration conditions, roughness conditions, or networks.

The comparison standardizes operators to

$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$0

uses maximum solution sizes of 15 and 30, and tests maximum numbers of view-specific parameters equal to 2 and 4 wherever the implementation permits it. Each configuration is repeated 10 times with different random seeds and 80/20 train/test splits. The main metric is a reduced MSE,

$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$1

with normalized targets $$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$2; in multi-view evaluation, the final loss is aggregated by $$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$3 across views (Russeil et al., 1 Sep 2025).

The runtime contrast is sharp. Across the five datasets, average runtimes are on the order of $$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$4–$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$5 seconds for the base Operon configuration, $$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$6–$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$7 seconds for an increased Operon setup, $$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$8–$$\mathcal{D}^{(i)}=\{(x_j^{(i)},y_j^{(i)})\}_{j=1}^{p_i},$$9 seconds for eggp, $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$0–$$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$1 seconds for PySR, and $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$2–$$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$3 seconds for PhySO (Russeil et al., 1 Sep 2025). Operon is therefore orders of magnitude faster, although the study also reports that its base configuration underperforms in model quality unless its evaluation budget is substantially increased.

In terms of solution quality, the comparison does not produce a single dominant method. eggp yields the largest number of satisfactory models under the reported MSE criteria; PhySO is similar and often produces physically coherent expressions; PySR produces fewer satisfactory models overall but achieves the highest number of best-performing models per dataset; and Operon, especially in its enlarged configuration, produces many accurate models but often over-parameterized ones (Russeil et al., 1 Sep 2025). The authors’ practical recommendation is not to rely on a single MvSR engine: different systems induce different structural biases, and exporting top-$$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$4 candidates is preferable to selecting one expression solely by scalar loss.

5. Regime-based MvSR in astronomical classification

The SDSS DR17 study provides a distinct but technically precise realization of MvSR: the views are not separate experiments but three contiguous redshift regimes of a single one-dimensional feature space $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$5 (Deshpande et al., 27 Feb 2026). The discovery subset contains 10,000 objects and is split into three equal-sized contiguous segments in $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$6, each further divided into 50% training and 50% validation. MvSR then learns one global symbolic structure $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$7 shared across views, with distinct parameter vectors $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$8 optimized per regime.

The symbolic search engine is GP with population size $$y^{(i)} \approx f(x^{(i)};\theta^{(i)}), \quad i=1,\dots,k.$$9, 35 generations, binary operators $f(x;\theta)$0, unary operator $f(x;\theta)$1, and terminals consisting of the single feature $f(x;\theta)$2 and scalar constants. Complexity is defined as node count,

$f(x;\theta)$3

with the hard constraint $f(x;\theta)$4. For a candidate structure, MvSR fits parameters independently in each view by GPU-accelerated L-BFGS and evaluates the template by the min–max aggregated loss

$f(x;\theta)$5

so that the fitness is determined by the worst-performing redshift regime (Deshpande et al., 27 Feb 2026).

The discovered representative scoring function is

$f(x;\theta)$6

a rational form with an exponential term in the denominator and total complexity 10. Discrete classes are then obtained by thresholding the continuous score,

$f(x;\theta)$7

Thresholds are selected by grid search maximizing Cohen’s $f(x;\theta)$8 in a five-fold stratified cross-validation protocol on 100,000 spectroscopically confirmed objects (Deshpande et al., 27 Feb 2026).

Under that protocol, MvSR is the top-performing symbolic method in the study, with

$f(x;\theta)$9

and thresholds

$\theta^{(i)}$0

The row-normalized confusion matrix gives correct-class probabilities of $\theta^{(i)}$1 for galaxies, $\theta^{(i)}$2 for stars, and $\theta^{(i)}$3 for QSOs; the reported $\theta^{(i)}$4 is approximately $\theta^{(i)}$5 and $\theta^{(i)}$6 approximately $\theta^{(i)}$7 (Deshpande et al., 27 Feb 2026). The same paper argues that the redshift-only task has an information-theoretic ceiling near $\theta^{(i)}$8, with MvSR essentially matching the Random Forest baseline and remaining only slightly below SVM and MLP on the same one-dimensional input (Deshpande et al., 27 Feb 2026).

This case study is important conceptually because it shows that MvSR need not require multiple feature sets or separate experiments. A regime-partitioned one-feature problem can also be cast in multi-view form when the aim is to enforce a structure that is uniformly adequate across distinct operating regions.

6. Broader interpretations, misconceptions, and open directions

Several adjacent SR frameworks enlarge the meaning of “multi-view” beyond the original multi-dataset setting. MMSR formulates symbolic regression as a purely multimodal information fusion task with a numeric view and a symbolic view, uses a SetTransformer for data encoding, a Transformer for skeleton encoding and decoding, and aligns the two views through an InfoNCE-style contrastive loss

$\theta^{(i)}$9

Its ablation study reports average $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$0 values of $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$1, $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$2, and $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$3 for $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$4, $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$5, and $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$6, respectively, and the paper states that MMSR achieves the most advanced results on multiple mainstream datasets including SRBench (Li et al., 2024).

ViSymRe extends the multimodal perspective further by integrating three modalities—vision, symbol and numeric—through canonical equation images, a real visual stream, a virtual visual stream, contrastive alignment, and a constrained Transformer decoder (Li et al., 2024). The reported joint objective is

$\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$7

The paper reports strong generalization capability and noise resistance, lower equation complexity at similar or better $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$8 than numeric-only baselines, and symbolic solution rates on Feynman problems that are 2–3× higher than most numeric-only baselines except AI Feynman (Li et al., 2024).

ScaleSR does not use the term MvSR, but its control-variable decomposition is explicitly described as inherently factorized and interpretable in multi-view terms. It learns a DNN generator, creates conditional data slices by controlling variables, applies single-variable SR inside those slices, and combines the resulting coefficient functions bottom-up. On multi-variable benchmarks its average recovery rate is $\min_{f,\theta}\mathcal{L}(f(x;\theta),y),$9, compared with $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$0 for SPL, $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$1 for NGGP, $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$2 for DSR, and $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$3 for GP (Chu et al., 2023). A plausible implication is that some future MvSR systems will combine shared symbolic templates with low-dimensional conditional subproblems rather than fitting all variables jointly.

At the representation level, IsalSR is not an MvSR algorithm but is explicitly presented as well suited to multi-view scenarios because it canonicalizes expression DAGs into a pruned canonical string, collapsing structurally redundant encodings and enabling cross-view equivalence detection (Lopez-Rubio et al., 23 Mar 2026). The reported empirical results include 100% canonical invariance across 961 tested DAGs and a local-neighborhood redundancy estimate of 71.2% around $$\min_f \; \mathrm{agg}_{i=1..k}\left(\min_{\theta^{(i)}} \mathcal{L}\big(f(x^{(i)};\theta^{(i)}),y^{(i)}\big)\right),$$4. This suggests that future MvSR systems may increasingly treat canonical structure, not only prediction loss, as a first-class optimization object.

The main open issues recur across the literature. The original MvSR paper identifies missing support for explicit parameter-count penalties and repeated parameters (Russeil et al., 2024). The physical-sciences comparison emphasizes parameter and constant control, uncertainty handling, support for weighted least squares, and extension beyond one-dimensional real-world datasets (Russeil et al., 1 Sep 2025). MMSR notes poor anti-noise performance, a fixed symbol set after training, and a fixed maximum number of variables (Li et al., 2024). ViSymRe highlights distribution shift in support-point ranges and constant scales on SRSD-Feynman (Li et al., 2024). Taken together, these papers indicate that the central unresolved problem is not whether shared-structure symbolic modeling across views is useful, but how best to balance expressivity, parameter economy, robustness, and scientific plausibility within a unified MvSR framework.