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DeSyR: A Decoupled Symbolic Recovery Framework with PINN-Guided Structure Search and Physics-Informed Coefficient Refinement

Pancheng Niu, Jun Guo, Qiaolin He, Jingcai Guo, Yanchao Shi

arXiv:2609.00530v1cs.LG

TL;DR

Recovering explicit symbolic solutions from imperfect neural approximations requires separating topology discovery from final coefficient estimation. DeSyR uses PINN-guided searches to propose candidate topologies, then performs physics-only coefficient refinement and gated verification. Across 15 problems and 18 configurations, the method achieved near-machine-precision results in many cases, while its guarantees and success depend on representability and convergence.

  • Problem

    Neural solution approximations may capture solution form while retaining coefficient errors and an implicit network representation, complicating compact explicit recovery.

  • Method

    DeSyR uses PINN-guided symbolic searches for candidate topologies and provisional coefficients, then refines frozen-topology coefficients solely from governing equations and prescribed constraints.

  • Results

    Across 15 differential-equation problems and 18 configurations, physics-only refinement reached near-machine-precision accuracy in many cases and reduced same-topology relative L2 error by eight to fourteen orders of magnitude.

  • Takeaways & Limitations

    Within tested representable settings, an approximate neural teacher can guide topology discovery without necessarily imposing its error scale on final recovered coefficients when a target-capable topology is retained and refinement converges.

  • Takeaways & Limitations

    Coefficient refinement cannot remove representation error when the target lies outside the prescribed symbolic space, and operator-library enrichment can reduce recovery under finite search budgets.

Abstract

from arXiv · show

Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an $O(β^{-1})$ teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative $L_2$ errors are $2.31\times10^{-14}$ or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.

1. Introduction

DeSyR addresses the difficulty of recovering compact explicit solutions from imperfect neural approximations by separating symbolic topology discovery from coefficient estimation. It combines PINN-guided candidate generation with physics-only refinement, supported by fixed-topology analysis and broad differential-equation experiments.

  • Motivation: Neural methods approximate continuous solution fields but typically leave solutions implicit in network parameters.PINN accuracy can be affected by optimization, loss imbalance, spectral bias, and stiff or multiscale problems.
  • Motivation: Symbolic regression searches combinatorial expression spaces whose admissible solution class is determined by the prescribed operator set.Existing neural-to-symbolic methods primarily estimate coefficients from the same teacher data used to guide structure search.
  • Framework: DeSyR assigns PINN outputs to topology search and provisional coefficients, then re-estimates fixed-topology coefficients solely from governing equations and prescribed constraints.Explicit reliability gates and verification residuals assess the resulting expressions.
  • Theory: For linear fixed-topology parameterizations, finite-weight mixed data–physics fitting retains an O(β^-1) teacher-dependent contribution when teacher error projects onto the model space.Under stated well-posedness, representability, zero-residual, and determinacy conditions, physics-only refinement recovers exact coefficients; nonlinear guarantees are local.
  • Evaluation: A 99.23% candidate-level convergence rate and configuration-level median refined relative L2 errors of 2.31×10^-14 or lower were obtained across 15 problems and 18 configurations.Same-topology refinement reduced errors by eight to fourteen orders of magnitude in cases with positive errors before and after refinement.

2. Problem formulation

The paper formulates recovery as finding a compact, differentiable explicit expression for a known forward differential problem under prescribed constraints. It separates admissible expression topology from numerical coefficients and evaluates candidates with reference-free physical-consistency measures plus optional post hoc error.

  • 2.1. Governing problem and scope: The target is a compact explicit representation of a solution field for a known governing equation and prescribed boundary or initial conditions.The formulation covers steady and time-dependent problems over domains in R^d.
  • 2.1. Governing problem and scope: The forward problem consists of a governing differential operator equation on Ω and constraint-operator equations on prescribed subsets Γc,ℓ.Constraint operators and the governing operator may be nonlinear, and constraints may be Dirichlet, Neumann, or initial conditions.
  • 2.1. Governing problem and scope: The analysis assumes the forward problem admits a unique classical solution u⋆ in the solution class under consideration.This is the paper’s stated well-posedness assumption.
  • 2.1. Governing problem and scope: The task is not governing-equation discovery or neural-solver improvement; the operator is known, and coupled fields are searched separately but refined and selected jointly.The objective is explicit-expression recovery using a fixed neural approximation and prescribed governing problem.
  • 2.2. Admissible symbolic representation: Expressions are built from a problem-specific operator library O that is fixed before and throughout symbolic search.The library acts as an ansatz specification and can create non-representability when it omits required operators.
  • 2.2. Admissible symbolic representation: Structural complexity is the total number of nodes in an expression tree, with raw search complexity distinguished from final post-refinement complexity.Raw complexity supports Pareto search, while final complexity supports Stage-C selection and reporting.
  • 2.2. Admissible symbolic representation: Exact symbolic recovery requires functional equality when the target lies in the bounded admissible space; otherwise the goal is a compact expression with small governing and constraint residuals.Admissibility also requires domain definition, required regularity, differentiability, and finite numerical evaluation on discrete point sets.
  • 2.3. Physical consistency and verification measures: Physical consistency is measured by normalized governing-equation and constraint residuals, while the unnormalized refinement objective is not interpreted as a verification measure.Reference solutions are used only for post hoc relative L2 error when available, not for search, selection, or refinement; dimensional applications require nondimensionalization or scaling.

3. The DeSyR framework

DeSyR uses a three-stage workflow: teacher-guided searches propose symbolic candidates, physics-only refinement re-estimates coefficients on frozen topologies, and explicit gates select and verify the final expression. Its operating conditions separate candidate coverage and teacher adequacy from refinement convergence.

  • Stage separation: Teacher samples propose candidate topologies, provisional coefficients, and compatibility assessments, while reference-solution values are excluded from the three recovery stages.The governing equation and prescribed constraints define the teacher-free refinement objective.
  • Stage B: refinement: Stage B freezes each candidate tree and re-estimates only its numerical constants with an objective containing governing-equation and constraint residuals but no teacher-data term.The retained numerical solution may come from a multi-start procedure, while the ideal minimizer is defined separately.
  • Stage A: search: Stage A pools up to five representative Pareto candidates from each of K independent symbolic searches.The pooled set combines search outputs before fixed-topology refinement.
  • Stage C: selection: Stage C applies convergence-eligibility, teacher-compatibility, physics-equivalence, and complexity gates to the refined candidate pool.Selection is performed over candidates rather than by a single mixed teacher–physics objective.
  • Operational conditions: The exact-recovery pathway requires teacher guidance adequacy, candidate-pool coverage, and finite converged fixed-topology refinement.These C1–C3 conditions are operational supports, not sufficient conditions for exact symbolic recovery.
  • PINN teacher: PINN training uses governing-equation and constraint residuals at collocation points, with fixed configuration weights and Adam followed by L-BFGS.The trained PINN supplies samples for Stage-A symbolic search.
  • Search–refinement decoupling: The search objective uses teacher fit and expression complexity without governing-equation or constraint residuals, so search-stage constants remain provisional.This preserves the separation between topology proposal and final coefficient determination.
  • Coupled systems: For coupled systems, fieldwise topology searches remain separate, while coefficient refinement and selection operate jointly under the shared governing system.The information separation across the three stages is unchanged.

4. Theoretical analysis

The theoretical analysis separates fixed-topology coefficient refinement from topology discovery, showing when physics-only refinement can recover exact coefficients and how teacher error affects alternative objectives. Linear guarantees are global under stated assumptions, whereas nonlinear guarantees are local and initialization-dependent.

  • Fixed-topology recovery: Fixed-topology refinement reduces symbolic recovery to estimating numerical coefficients after the expression tree is frozen.The representation component cannot be removed by coefficient optimization when the topology is not expressive enough.
  • Linear error inheritance: Teacher fitting inherits coefficient error whenever teacher error projects onto the sampled model space, with amplification governed by 1/σmin(Φ).This bias need not vanish with increasing sample size unless the teacher error becomes asymptotically orthogonal to the model space.
  • Conditional exact recovery: Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers the exact coefficients and solution on a fixed topology.The zero-residual target contains no teacher quantity once the topology, parameterization, and refinement point set are fixed.
  • Initialization dependence: Teacher information affects fixed-topology nonlinear optimization through initialization, influencing basin of attraction and iteration cost but not the zero-residual target when attained.The result does not establish topology-discovery guarantees or global nonlinear convergence.
  • Finite-weight mixed objectives: For every finite β > 0, mixed data–physics fitting retains nonzero teacher-dependent coefficient bias when Φ⊤ε ≠ 0.The bias decreases toward the physics-only solution as β increases, but the stated asymptotic does not imply monotonic error decay.
  • Nonlinear local analysis: In nonlinear parameterizations, identifiability and convergence guarantees are local: convergence is guaranteed only for initializations inside a neighborhood of the target.Outside that neighborhood, spurious stationary points may be reached, and the linear global bounds do not apply.

5. Numerical experiments

Across the benchmark suite, fixed-topology physics-only refinement consistently produced highly accurate expressions, including for nonlinear, coupled, and difficult high-frequency cases. Additional audits show strong candidate convergence and identify representational, initialization, and candidate-coverage boundaries.

  • All 90 selected expressions were recorded as converged, and every selected refinement involving free coefficients converged successfully.
  • Refinement reduced the Burgers median relative L2 error from 3.92 × 10−4 before refinement to 3.71 × 10−17 afterward while preserving the travelling-wave topology.
  • The Helmholtz candidate preserved the high-frequency product structure and reached a median relative L2 error of 2.31 × 10−14 after coefficient refinement.The result demonstrates improvement over the teacher once a suitable symbolic structure has been identified.
  • Joint Kovasznay recovery achieved median relative L2 errors of 1.81 × 10−15 for u, 3.40 × 10−15 for v, and 3.69 × 10−15 for p.The recovered fields captured the coupled exponential and trigonometric structure while enforcing incompressibility.
  • Pooling a second independently seeded search recovered all 60 instances, but the archived protocol cannot distinguish ungenerated target topologies from candidates removed by five-candidate truncation.The result is therefore an end-to-end candidate-coverage indicator rather than a direct measure of algebraic topology coverage.
  • Fixed-topology refinement corrected representative coefficient mismatches by approximately ten orders of magnitude while preserving the search-stage topology.For Sine–Poisson 1D, α moved from 3.1415536 to 3.14159265358979 and the relative L2 error fell from 2.48 × 10−5 to 1.98 × 10−15.
  • As β increased, linear mixed estimates approached the physics-only solution with estimated tail slopes close to −1, whereas nonlinear cases showed the same progression without an assigned asymptotic rate.The physics-only endpoint in the reported linear comparison reached 1.86 × 10−17.

6. Discussion

DeSyR separates teacher-guided topology discovery from physics-only coefficient refinement, with verification exposing representation mismatch and diagnostics testing convergence and coverage conditions.

  • Discussion: DeSyR uses PINN outputs for topology search but determines final coefficients from governing physics and prescribed constraints after topology fixation.This differs from teacher-only and mixed data–physics coefficient objectives.
  • Discussion: Successful recovery requires informative teacher guidance, a target-capable retained topology, and a finite, converged refinement.These operational conditions separate teacher guidance, candidate coverage, and Stage-B convergence.
  • Discussion: Candidate-level audits distinguish convergence of all Stage-B refits from convergence of the selected expression, while enrichment tests indicate larger libraries may require larger search budgets.These diagnostics provide empirical support for candidate coverage and refinement-convergence conditions under tested finite-budget settings.
  • Discussion: Theoretical guarantees concern fixed structures; nonlinear identifiability and convergence are local, and topology-discovery or global nonlinear convergence is not established.The study also focuses on controlled benchmarks with prescribed libraries, regular domains, and analytic reference solutions.
  • Discussion: Future extensions must reassess coverage, refinement robustness, and verification for noisy observations, irregular geometries, discontinuous or multiscale solutions, and richer symbolic spaces.Adaptive operator screening or hierarchical library expansion is proposed to improve coverage while controlling combinatorial growth.
  • Discussion: Constraint perturbation refits all converged, but increasing perturbation raised error relative to the unperturbed reference and did not establish general stability.The study reports 48 converged fixed-topology refits under perturbed constraints.

7. Conclusions

DeSyR decouples symbolic topology discovery from coefficient determination: a PINN proposes structures, while physics-only refinement recovers coefficients under stated conditions. Across broad benchmarks, refinement often reaches near-machine precision, but representation mismatch and finite search budgets remain boundaries.

  • 7. Conclusions: DeSyR separates PINN-guided topology discovery from teacher-free coefficient refinement using governing equations and prescribed constraints.The final refinement objective excludes teacher samples once topology and refinement points are fixed.
  • 7. Conclusions: Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients.For nonlinear parameterizations, the corresponding identifiability and convergence guarantees are local and initialization-dependent.
  • 7. Conclusions: Eight to fourteen orders of magnitude: same-topology relative L2 error reductions occur across reported comparisons with strictly positive prerefinement and post-refinement errors.The evaluation spans 15 differential-equation problems and 18 configurations, including nonlinear and coupled cases.
  • 7. Conclusions: An approximate neural teacher need not impose its error scale on recovered coefficients when a target-capable topology is retained and physics-only refinement converges.This conclusion is supported within the tested representable settings.

Appendix A. Proofs of theoretical results

The proofs establish fixed-topology coefficient behavior through least-squares identities, residual objectives, injectivity, and local Hessian arguments. They connect zero residual and local regularity to exact recovery or local convergence under explicit assumptions.

  • Appendix A. Proofs of theoretical results: Least-squares coefficients satisfy normal equations derived from a strictly convex quadratic when the feature Gram matrix is positive definite.The coefficient solution uses the inverse feature Gram matrix and its smallest singular value controls perturbation amplification.
  • Appendix A. Proofs of theoretical results: Residual decomposition bounds recovered-solution error by separating coefficient perturbation from representation error around the exact or reference parameterization.The proof applies the triangle inequality and submultiplicativity to the linear parameterization.
  • Appendix A. Proofs of theoretical results: Zero refinement objective implies zero residual, and injectivity near the exact coefficient vector then forces equality with the exact coefficients.The refinement objective uses only governing operators, constraints, forcing, points, and weights, not teacher samples or reference values.
  • Appendix A. Proofs of theoretical results: Teacher dependence in Stage B is conditional on initialization once topology, parameterization, and refinement points are fixed.The refinement objective itself contains no teacher-dependent quantity.
  • Appendix A. Proofs of theoretical results: Finite-weight mixed data–physics fitting remains strictly convex for positive weights when both data and physics Gram matrices are positive definite.The resulting linear system combines Φ^TΦ with βA^TA, and its asymptotics are analyzed through matrix inversion and a Neumann expansion.
  • Appendix A. Proofs of theoretical results: A positive-definite Hessian near a zero-residual solution yields strict local minimality and uniqueness within a neighborhood.Local Hessian regularity then supports quadratic Newton convergence; Gauss–Newton matches the exact Hessian at the zero-residual solution.

Appendix B. Benchmark definitions and fixed configurations

Appendix B defines 18 benchmark configurations by their equations, domains, constraints, reference solutions, operators, and fixed numerical settings, with complete search settings provided separately.

  • Appendix B. Benchmark definitions and fixed configurations: The governing equations, domains, constraints, reference solutions, and unary operators for all 18 configurations are summarized in Table B.1.Table B.1 serves as the appendix’s benchmark-definition index.
  • Appendix B. Benchmark definitions and fixed configurations: 18 configurations are specified by governing equations, computational domains, prescribed constraints, reference solutions, and problem-specific unary operators.The table defines u⋆ as the analytic reference solution where applicable and identifies ∂Ωx as the spatial boundary for time-dependent problems.
  • Appendix B. Benchmark definitions and fixed configurations: The Kovasznay benchmark represents u = (u, v) as a velocity field.This notation is part of the benchmark-definition table.
  • Appendix B. Benchmark definitions and fixed configurations: Complete search-operator settings are reported in Table B.3.The appendix points readers to the table for the full operator configuration.
  • Appendix B. Benchmark definitions and fixed configurations: Fixed numerical settings are consolidated by configuration ID in Tables B.2 and B.3.These settings complement the benchmark definitions and search-operator specifications.

Appendix B.1. PINN and sampling settings

The appendix specifies PINN architectures, sampling conventions, and the distinct point sets used for equation residuals and coefficient refinement.

  • PINN architectures and sampling settings cover 18 configurations, with training-point counts reported as interior/boundary/initial locations.A dash indicates an unused point class, and counts refer to sampled locations rather than scalar residual components.
  • Physics validation and Stage-B coefficient refinement use separate interior equation-residual point sets, while constraint terms use separately prescribed constraint point sets.

Appendix B.2. Symbolic-search and refinement settings

The symbolic pipeline uses repeated seeded searches, candidate-pool ranking, specialized shared-factor processing, and subsequent coefficient refinement settings.

  • Ten independently seeded symbolic-search runs per PINN teacher retain up to five representative candidates through Stage-A Pareto-front selection.Configuration-specific refinement-point counts are provided separately, with additional coupled-search settings for Kovasznay.
  • Kovasznay candidate groups combine fieldwise shortlists of at most 12 candidates per field and rank joint groups before screening refinement.The 16 highest-ranked groups undergo joint screening refinement using one initialization.
  • Shared-factor augmentation scans single-variable exponential factors with rates |r| ≥ 10^-10 and processes observations in increasing absolute-rate order.
  • Eligible cross-field rate clusters require support from at least two fields, after which the first two are retained using support, loss, dispersion, and rate criteria.Within each field, up to three trigonometric factors are selected using candidate loss, expression-tree size, and symbolic string ordering.
  • Kovasznay joint coefficient sharing represents matched exponential rates as integer multiples of shared base parameters when k ∈ {1, 2, 3, 4} and the ratio tolerance is satisfied.Rates matching no shared base remain field-specific.

Appendix C. Detailed numerical results

Appendix C supplements the main configuration-level medians with across-seed distributions and clarifies how errors, complexity, and exact zeros are reported.

  • Configuration-level medians are reported in Table 4, while the consolidated appendix table adds across-seed dispersion without repeating the 18 configurations in multiple class-specific layouts.
  • Across-seed relative L2 errors are reported as median [first quartile, third quartile] over five independently initialized PINN teachers.The pre-refit expression is the search-stage expression corresponding to the candidate ultimately selected after refinement and Stage-C selection.
  • The complexity column reports the range of final expression complexity across the five seeds.
  • Exact floating-point zeros are stored without applying a display floor.

Appendix C.1. Candidate-pool coverage under repeated searches

The appendix evaluates end-to-end recovery across repeated teacher instances and audits whether prescribed Helmholtz target-family candidates survive candidate retention.

  • Separately seeded symbolic searches enlarge the candidate pool for a fixed teacher but are not additional independent replicates.
  • Each first-K replay entry counts successful end-to-end recoveries among five independently trained PINN teachers, with success defined by relative L2 error no larger than 10^-10.The last column reports the smallest tested K yielding recovery for all five teacher-level instances.
  • Raw Pareto-front and retained-pool counts coincide across all three Helmholtz library-enrichment settings for candidates matching the prescribed target family.The audit may undercount equivalent expressions that require coefficient refinement and symbolic simplification.
  • The Helmholtz structural audit counts teacher instances containing at least one candidate of the family C sin(ax + b) sin(cy + d) + e.

Appendix C.2. Coefficient-refinement objective ablation

The appendix compares coefficient-refinement objectives and initialization choices for fixed, linear-capable topologies, using error and residual diagnostics across problem–seed combinations.

  • Each linear fixed-topology basis admits an exact solution representation and is parameterized linearly in its coefficients.
  • The fixed-topology experiments include Multifrequency Poisson and Euler–Bernoulli bases with three and four coefficients, respectively.
  • The objective ablation evaluates relative L2 errors and equation residuals across 50 problem–seed combinations from ten scalar configurations with free coefficients.
  • The appendix separately reports exact floating-point zeros without applying a display floor.
  • Initialization sensitivity compares one teacher-derived initialization with five independent random initializations for each of five PINN seeds per configuration.

Appendix C.4. Sensitivity to Perturbations in Prescribed Constraints

This appendix tests refinement under perturbed prescribed constraints and supplies grouped diagnostics for recovery quality, complexity, timing, and Stage-C selection behavior.

  • Constraint perturbation sensitivity: The perturbation study changes only prescribed boundary or initial values; the governing equation remains unperturbed and the topology is fixed to a target-capable form.
  • Constraint perturbation sensitivity: All 48 refits converged, while constraint noise generally increased both solution error and governing-equation residual.At η = 10^-2, median relative L2 errors were 6.20 × 10^-4 for Multifrequency Poisson, 3.68 × 10^-7 for Helmholtz, and 7.20 × 10^-4 for Burgers.
  • Recovery diagnostics: Representative profiles and field visualizations compare reference solutions with PINN teachers and refined DeSyR expressions using shared logarithmic error scales.
  • Recovery diagnostics: Seed-level diagnostics report PINN, pre-refit, and refined relative L2 errors separately across five seeds for all 18 configurations.
  • Computational and selection diagnostics: Configuration-level timing summaries separate PINN training from symbolic recovery, which includes search, refinement, Stage-C selection, and verification.
  • Computational and selection diagnostics: Removing convergence eligibility admits one unconverged candidate and substantially worsens the worst selected error, whereas removing complexity preference changes 12 selections without increasing it.
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