Sym2Real: Bridging Symbolic and Real Domains

• Sym2Real is a framework that relates structured source domains—symmetric, symbolic, or simulation-calibrated—to real target domains via explicit maps and correspondences.
• In geometric contexts, it employs invariant and equivariant mappings to establish homeomorphisms between real and complex or symmetric moduli, enhancing analysis in Lie groups and matrix loci.
• In robotics and control, the two-stage framework uses symbolic regression on low-fidelity simulations followed by real-data residual learning to achieve efficient sim-to-real transfer.

Searching arXiv for recent and foundational uses of “Sym2Real” and closely related papers mentioned in the data. Sym2Real denotes several research programs in which a structured source domain—typically symmetric, symbolic, or simulation-calibrated—is related to a real target domain by a map, correspondence, or adaptation procedure. In geometric representation theory, the term is associated with correspondences between real moduli attached to a real reductive group and complex algebraic moduli attached to its symmetric variety, with homeomorphisms between real and symmetric matrix loci, and with effective descent from complex symmetric varieties to models defined over real algebraic numbers (Chen et al., 2018, Chen et al., 2020, Hidalgo, 2012). In robotics and control, Sym2Real names a two-stage framework that learns symbolic dynamics in low-fidelity simulation and specializes them with a small real-world residual, while adjacent Real2Sim2Real systems calibrate simulation from real data before transferring policies or data back to reality (Lee et al., 18 Sep 2025, Han et al., 12 Feb 2025).

1. Representation-geometric Sym2Real correspondences

A central geometric formulation begins with a connected complex reductive group $G$, a real form $G_\mathbb R \subset G$ given by a conjugation $\eta$, a Cartan conjugation $\delta$ commuting with $\eta$, the Cartan involution $\theta=\delta\circ\eta=\eta\circ\delta$, and the fixed-point subgroup $K=G^\theta$. The associated complex symmetric variety is

$X:=K\backslash G.$

The Cartan bijection is realized by

$\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$

which descends to an isomorphism $X\simeq G_{\mathrm{sym}}^0$, the neutral component of

$G_\mathbb R \subset G$0

Within this setup, constructions depending on the real form $G_\mathbb R \subset G$1 can be transferred to the symmetric space determined by $G_\mathbb R \subset G$2 (Chen et al., 2018).

An analogous matrix-theoretic realization is provided on the real-spectrum locus

$G_\mathbb R \subset G$3

Here one constructs a family of involutions $G_\mathbb R \subset G$4, $G_\mathbb R \subset G$5, interpolating

$G_\mathbb R \subset G$6

and deduces an $G_\mathbb R \subset G$7-equivariant stratified homeomorphism

$G_\mathbb R \subset G$8

which restricts to a real-analytic isomorphism between individual $G_\mathbb R \subset G$9-adjoint orbits and $\eta$0-adjoint orbits. The same framework extends to Lie algebras of classical types and quiver varieties, and is used for applications to the generalized Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces (Chen et al., 2020).

These constructions establish a precise sense in which real and symmetric loci are locally or stratifiably equivalent without identifying them as the same algebraic object. The preserved data are topological and orbit-theoretic: spectrum, adjoint quotient, and complex orbit type are retained, while the endpoint involutions interpolate between conjugation and transpose or, more generally, between real and symmetric fixed-point loci.

2. Arcs, quasi-maps, and the Sym→Real principle

For a pointed variety $\eta$1, the based polynomial arc space is

$\eta$2

For a group $\eta$3, one writes

$\eta$4

In the global-curve convention used for $\eta$5 and $\eta$6, the open cospherical cell $\eta$7 in the real affine Grassmannian is identified with

$\eta$8

and similarly

$\eta$9

The main arc-space theorem is the existence of a $\delta$0-equivariant stratified homeomorphism

$\delta$1

real analytic on strata (Chen et al., 2018).

The multi-point version is formulated over $\delta$2. For marked points $\delta$3, the fibers are

$\delta$4

and

$\delta$5

Stratification is by modification type using a partition $\delta$6 of $\delta$7 and a map $\delta$8. The resulting spherical strata $\delta$9 form a Whitney stratification compatible with the parameter stratification by point collisions, and the multi-point theorem gives a $\eta$0-equivariant stratified homeomorphism fibered over $\eta$1 that restricts to real-analytic isomorphisms on these strata.

The quasi-map formulation replaces arcs by moduli of quasi-maps from $\eta$2. For a target $\eta$3, one has the moduli space $\eta$4 of quasi-maps $\eta$5 with allowed poles at the marked points $\eta$6, together with a based version $\eta$7 after rigidification at a basepoint $\eta$8. These moduli spaces are uniformized by Beilinson-Drinfeld Grassmannians, and the main multi-point correspondence becomes a $\eta$9-equivariant stratified homeomorphism

$\theta=\delta\circ\eta=\eta\circ\delta$0

compatible with evaluation at marked points and with spherical stratification by modification types.

Two technical devices drive the construction. The first is a multi-point generalization of the classical polynomial loop factorization

$\theta=\delta\circ\eta=\eta\circ\delta$1

upgraded to punctured projective lines with marked real and conjugate points. The second is a nodal degeneration $\theta=\delta\circ\eta=\eta\circ\delta$2 with fibers $\theta=\delta\circ\eta=\eta\circ\delta$3 for $\theta=\delta\circ\eta=\eta\circ\delta$4 and central fiber $\theta=\delta\circ\eta=\eta\circ\delta$5 a nodal curve given locally by $\theta=\delta\circ\eta=\eta\circ\delta$6. The twisted conjugation

$\theta=\delta\circ\eta=\eta\circ\delta$7

exchanges the components at $\theta=\delta\circ\eta=\eta\circ\delta$8, and because the real locus of $\theta=\delta\circ\eta=\eta\circ\delta$9 is only the node, the real matching conditions on the generic fiber degenerate into purely complex algebraic matching conditions at the node. This yields the paper’s “Sym→Real principle”: local real moduli for $K=G^\theta$0 are $K=G^\theta$1-equivariantly, stratifiedly homeomorphic to complex algebraic moduli for $K=G^\theta$2, and the singularities of closures of real spherical orbits in the real affine Grassmannian are locally homeomorphic to singularities in complex algebraic varieties (Chen et al., 2018).

3. Effective descent from symmetric varieties to real models

A different Sym2Real formulation occurs in effective descent for arithmetical real algebraic varieties. Let $K=G^\theta$3 be a smooth complex algebraic variety admitting a symmetry $K=G^\theta$4, meaning an antiholomorphic automorphism of order two, and let $K=G^\theta$5 denote coordinate-wise complex conjugation. The framework assumes that $K=G^\theta$6 and $K=G^\theta$7 are defined over a conjugate-invariant subfield $K=G^\theta$8, that $K=G^\theta$9 is biregular, and that one seeks a model over $X:=K\backslash G.$0. The main theorem gives an explicit rational map $X:=K\backslash G.$1, defined over $X:=K\backslash G.$2, such that $X:=K\backslash G.$3 is defined over $X:=K\backslash G.$4 and

$X:=K\backslash G.$5

is the standard conjugation on $X:=K\backslash G.$6 (Hidalgo, 2012).

The construction proceeds in two steps. First, writing $X:=K\backslash G.$7, one defines

$X:=K\backslash G.$8

Second, one forms a polynomial map $X:=K\backslash G.$9, where $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$0, from the degree-$\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$1 invariants of the $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$2-action swapping the two factors: $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$3 The descent map is then

$\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$4

If $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$5 is polynomial, $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$6 is biregular; in general, it is birational. The target involution is the usual coordinate-wise conjugation $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$7, and the paper gives an algorithmic elimination procedure, via the ideal generated by the equations of $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$8, the equations defining $\pi:G\to G,\qquad \pi(g):=\theta(g)^{-1}g,$9, and the relations defining the $X\simeq G_{\mathrm{sym}}^0$0-coordinates, to compute defining equations for $X\simeq G_{\mathrm{sym}}^0$1.

This construction is an effective version of the existential descent established by Koeck, Lau, and Singerman via Weil descent. The distinctive feature is explicitness: the image coordinates are written down, the descended model is exhibited over $X\simeq G_{\mathrm{sym}}^0$2, and an inverse $X\simeq G_{\mathrm{sym}}^0$3 can be recovered birationally from the invariant coordinates. In the language of the paper, the $X\simeq G_{\mathrm{sym}}^0$4-cocycle determined by the symmetry is trivialized after passing to invariant quadratic coordinates, turning the transported involution into ordinary conjugation (Hidalgo, 2012).

4. Symmetric real algebraic geometry and orbit-type restrictions

In projective geometry, symmetry imposes strong restrictions on the reality of intersection loci. For $X\simeq G_{\mathrm{sym}}^0$5 with the $X\simeq G_{\mathrm{sym}}^0$6-action by coordinate permutation, a symmetric hypersurface is the zero locus of a homogeneous polynomial invariant under that action. In $X\simeq G_{\mathrm{sym}}^0$7, combining equivariant conservation of number with Bézout yields a “symmetric Bézout theorem”: for symmetric curves of degrees $X\simeq G_{\mathrm{sym}}^0$8 and $X\simeq G_{\mathrm{sym}}^0$9 in general position, the $G_\mathbb R \subset G$00-isomorphism type of the finite intersection $G_\mathbb R \subset G$01 depends only on $G_\mathbb R \subset G$02. A complete classification is obtained modulo $G_\mathbb R \subset G$03 (Lidz et al., 2024).

$G_\mathbb R \subset G$04	Orbit type in $G_\mathbb R \subset G$05
$G_\mathbb R \subset G$06	$G_\mathbb R \subset G$07
$G_\mathbb R \subset G$08	transverse intersection not possible
$G_\mathbb R \subset G$09	$G_\mathbb R \subset G$10
$G_\mathbb R \subset G$11	$G_\mathbb R \subset G$12
$G_\mathbb R \subset G$13	transverse intersection not possible
$G_\mathbb R \subset G$14	$G_\mathbb R \subset G$15

The proof uses fixed-point analysis and tangent-line obstructions. If a symmetric polynomial vanishes at $G_\mathbb R \subset G$16, then that point is singular, so no $G_\mathbb R \subset G$17-fixed point occurs in a transverse intersection. In $G_\mathbb R \subset G$18, points of the form $G_\mathbb R \subset G$19 and $G_\mathbb R \subset G$20 force prescribed tangent lines, so two symmetric curves meeting there cannot intersect transversely. Consequently, at most one orbit of type $G_\mathbb R \subset G$21 and at most one orbit of type $G_\mathbb R \subset G$22 can occur.

Because the $G_\mathbb R \subset G$23-action commutes with complex conjugation, each orbit is either all real or all complex. In $G_\mathbb R \subset G$24, $G_\mathbb R \subset G$25-orbits contribute exactly $G_\mathbb R \subset G$26 real points, $G_\mathbb R \subset G$27-orbits contribute no real points, and an $G_\mathbb R \subset G$28-orbit contributes either $G_\mathbb R \subset G$29 real points or $G_\mathbb R \subset G$30. Hence, if $G_\mathbb R \subset G$31, the intersection of two real transverse symmetric curves has $G_\mathbb R \subset G$32 real points, while if $G_\mathbb R \subset G$33, it has $G_\mathbb R \subset G$34 real points. In $G_\mathbb R \subset G$35, the partial classification of admissible isotropy types implies that the product of the three degrees in a transverse intersection of symmetric surfaces is congruent to $G_\mathbb R \subset G$36, $G_\mathbb R \subset G$37, $G_\mathbb R \subset G$38, or $G_\mathbb R \subset G$39 modulo $G_\mathbb R \subset G$40, and any transverse intersection of three real symmetric surfaces has a number of real points divisible by $G_\mathbb R \subset G$41 (Lidz et al., 2024).

A computationally distinct but related use of symmetry appears in symbolic computation for real algebraic geometry. For the $G_\mathbb R \subset G$42-action on $G_\mathbb R \subset G$43, the invariant ring is

$G_\mathbb R \subset G$44

and Newton’s identities connect the elementary symmetric polynomials $G_\mathbb R \subset G$45 and power sums $G_\mathbb R \subset G$46. The chapter emphasizes a “Sym2Real pipeline” based on restriction to the Weyl chamber

$G_\mathbb R \subset G$47

use of Vandermonde maps, the half-degree principle, Reynolds averaging, and block-diagonalization of SOS Gram matrices. For symmetric optimization and decision problems, this reduces variable count, restricts attention to points with few distinct coordinates, and yields SDP sizes that stabilize for fixed degree once $G_\mathbb R \subset G$48 (Riener et al., 31 Jul 2025).

5. Sym2Real in adaptive control and symbolic regression

In control, “Sym2Real” is the name of a fully data-driven, two-stage framework for adaptive low-level control. The model class is a discrete single-step dynamics map

$G_\mathbb R \subset G$49

embedded in a sampling-based MPC controller using MPPI. Stage I learns a compact symbolic dynamics model from a deliberately low-fidelity simulator with default robot description files, noiseless observations, instantaneous control, and no domain randomization or disturbance injection. Stage II freezes this symbolic base and fits a small residual MLP on a few real trajectories: $G_\mathbb R \subset G$50 with regularized objective

$G_\mathbb R \subset G$51

The paper reports robust control of both a Crazyflie 2.1 quadrotor and a MuSHR racecar using about $G_\mathbb R \subset G$52 trajectories in total for deployment and adaptation, consistent data-efficient adaptation across six out-of-distribution sim2sim scenarios, and successful sim2real transfer across five real-world conditions (Lee et al., 18 Sep 2025).

The symbolic-regression stage uses PySR with operators $G_\mathbb R \subset G$53, expression length capped at $G_\mathbb R \subset G$54, five search iterations, and an $G_\mathbb R \subset G$55 loss. The MPC cost is task-specific; for the quadrotor it includes position, quaternion-based orientation, linear velocity, and angular velocity penalties, with position tracking error

$G_\mathbb R \subset G$56

The paper’s empirical claim is that symbolic regression extracts shared core physics efficiently from low-fidelity simulation, while the residual captures mass offsets, wind, friction, latencies, and imperfect thrust mapping. By contrast, directly fitting symbolic regression on real data produced physically wrong expressions, and a model-free RL controller trained in the same simplified simulation crashed the drone upon zero-shot deployment.

A precursor in symbolic-regression methodology reformulates symbolic regression itself as a smooth real-valued optimization problem by fixing an expression-tree structure and replacing discrete operator and variable choices by continuous weights. Internal nodes use weighted operator mixtures, leaves use weighted variable-and-constant selection, and custom penalties drive the weights toward crisp selections; the paper optimizes the resulting objective with CMA-ES. On Poly-10, the transformed landscape exhibits very high autocorrelation in the smooth space, and the stated goal is to make symbolic regression amenable to classical fitness-landscape analysis and continuous optimization rather than abrupt combinatorial edits (Pitzer et al., 2021).

6. Real2Sim2Real extensions in robotics and simulation

Adjacent robotics literature extends the transfer logic from symbolic or symmetric priors to calibrated simulation. Re$G_\mathbb R \subset G$57Sim is a 3D-photorealistic real-to-sim system for robotic manipulation that reconstructs the background mesh via COLMAP and OpenMVS, reconstructs foreground objects via ARCode, renders the background with 3D Gaussian Splatting and the foreground with mesh-based ray tracing, aligns simulator and world coordinates using ArUco markers and ICP, and then collects privileged expert demonstrations in simulation for imitation learning. Policies trained exclusively on simulated demonstrations achieve zero-shot transfer without real fine-tuning, with an average real success rate exceeding $G_\mathbb R \subset G$58 across three tabletop tasks (Han et al., 12 Feb 2025).

For deformable linear object manipulation, a distributional Real2Sim2Real pipeline treats adaptation as posterior inference over object parameters $G_\mathbb R \subset G$59, using BayesSim with RKHS embeddings to estimate

$G_\mathbb R \subset G$60

and then trains PPO policies by sampling simulator parameters from the learned object-specific posterior rather than from a uniform randomization range. The real system uses only monocular RGB vision and proprioception, and zero-shot deployment across $G_\mathbb R \subset G$61 real trials is used to analyze object-centric adaptation in end-effector trajectories (Kamaras et al., 25 Feb 2025).

In autonomous driving, SynthDrive explicitly formulates a “calibrate on real $G_\mathbb R \subset G$62 simulate $G_\mathbb R \subset G$63 transfer back to real” loop. The pipeline mines rare assets from real and web data with CLIP-guided retrieval, reconstructs textured 3D meshes from a single image using Zero123++, InstantMesh, StableNormal, and differentiable mesh refinement, inserts those assets into real driving videos or reconstructed scenes with HDR lighting estimation and occlusion-aware composition, and re-integrates synthetic data into training on real benchmarks. The paper reports over $G_\mathbb R \subset G$64 high-quality 3D assets, recommends mixing at roughly $G_\mathbb R \subset G$65 synthetic for the most consistent gains, and emphasizes realism controls rather than explicit closed-form camera or LiDAR sensor models (Chen et al., 8 Sep 2025).

Taken collectively, these usages suggest a family resemblance rather than a single formal doctrine. In the geometric literature, Sym2Real is a correspondence between real loci and symmetric or complex algebraic models. In control and robotics, it is a strategy for extracting structured dynamics or calibrated simulators from simplified domains and then specializing them with a limited amount of real data. The common pattern is the transfer of a compressed or symmetry-constrained representation—Cartan data, invariant coordinates, symbolic equations, posterior parameter distributions, or photorealistic digital twins—into a real deployment or real-moduli setting, with the claim that local topology, control behavior, or downstream performance can then be analyzed in the structured surrogate domain before being carried back to reality (Chen et al., 2018, Lee et al., 18 Sep 2025).