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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 compact explicit solutions from imperfect neural approximations is challenging when teacher errors affect coefficient estimation. DeSyR uses a PINN to guide topology search, then refines coefficients using only the governing equation and constraints. Across 15 problems and 18 configurations, same-topology refinement reduces relative L2 error by eight to fourteen orders of magnitude in reported comparisons.

  • Problem

    Neural solutions may capture the overall form yet retain coefficient errors and remain implicit, complicating recovery of compact explicit expressions.

    Source: §1 · pp. 2–3

  • Method

    DeSyR uses a PINN to guide symbolic topology search, then freezes the selected topology and refines eligible coefficients using only the equation and prescribed constraints.

    Source: §1 · p. 4, §7 · p. 57

  • Results

    Eight to fourteen orders of magnitude lower relative L2 error in positive-error same-topology comparisons; across 15 problems and 18 configurations, refinement reaches near-machine precision in many cases.

    Source: §1 · p. 5, §7 · p. 58

  • Takeaways & Limitations

    Within tested representable settings, 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.

    Source: §7 · p. 58

    Coefficient refinement cannot remove representation error when the target lies outside the prescribed symbolic space.

    Source: §5.4.5 · p. 52

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 uses PINN-generated candidates to discover symbolic topologies, then refines fixed-topology coefficients using only governing equations and constraints. Across its benchmark problems, physics-only refinement achieved very low errors, with theoretical guarantees conditional on stated assumptions.

Motivation

  • Neural methods can provide continuous but implicit solutions with errors in amplitudes or other coefficients, while symbolic regression can inherit teacher errors in fitted coefficients despite recovering the functional structure.Compact explicit solutions can facilitate interpretation, differentiation, and reuse; DeSyR addresses the gap by assigning teachers to topology search and physics to fixed-topology coefficient estimation.Source: §1 · pp. 2–4

Related work

  • Existing approaches fit expressions to teacher samples, jointly optimize structure and coefficients with data and physics, or partly separate these stages while retaining mixed search fitness.The introduction identifies incomplete separation of topology discovery from coefficient estimation, and a lack of systematic conditions for exact fixed-topology recovery, as the motivating gap.Source: §1 · pp. 3–4

Framework

  • DeSyR uses PINN outputs to guide repeated symbolic searches and initialize candidate coefficients, then freezes a selected topology and refines eligible coefficients solely from governing equations and prescribed constraints.Gated selection and verification residuals assess the resulting explicit expression, keeping teacher-data fitting out of the final refinement objective.Source: §1 · p. 4

Theory

  • For fixed linear topologies, finite-weight mixed data–physics fitting retains an O(β^-1) teacher-error contribution when that error projects onto the model space; physics-only exact recovery is conditional on stated assumptions.For nonlinear parameterizations, guarantees are local and depend on identifiability, conditioning, and initialization; they do not guarantee topology discovery or global convergence.Source: §1 · pp. 4–5

Results

  • Across 15 problems and 18 configurations, free-parameter refits converged at 99.23%, median refined relative L2 errors were at most 2.31×10^-14, and same-topology errors fell by 8–14 orders of magnitude.Every selected refinement with free coefficients converged; the reported order-of-magnitude reductions apply to cases with nonzero errors both before and after refinement.Source: §1 · p. 5

2. Problem formulation

DeSyR formulates symbolic recovery as finding a compact explicit solution to a known differential problem, using a fixed admissible expression space and physical-consistency measures. The formulation distinguishes topology from coefficients and separates reference-free verification from post hoc error reporting.

2.1. Governing problem and scope

  • Given a known governing operator and prescribed constraints, recovery seeks a compact explicit solution; under Assumption 2.1, the problem has a unique classical solution u⋆.The problem is written as N[u](x) = f(x) on Ω and Bℓ[u](x) = gℓ(x) on each constraint set Γc,ℓ.Source: §2.1 · pp. 5–6
  • The task assumes the differential operator is known, does not improve the neural solver, and searches coupled fields separately before joint refinement and selection.The paper focuses on scalar fields in this section.Source: §2.1 · p. 6

2.2. Admissible symbolic representation

  • The operator library is specified before search and remains fixed; admissible trees must be defined on Ω and sufficiently regular for the governing and constraint operators.Finite-value screening on discrete point sets is necessary numerically but does not establish regularity over the full domain.Source: §2.2 · pp. 6–7
  • Expression complexity counts tree nodes, distinguishing raw search complexity from final complexity after refinement and accepted simplification.Raw complexity guides Pareto search; final complexity supports selection and reporting.Source: §2.2 · p. 7
  • A topology fixes the expression tree but not constant values, which form a coefficient vector re-estimated while that topology remains fixed.Source: §2.2 · p. 7
  • If the solution lies in the admissible space, the target is functional equality; otherwise, recovery seeks a compact admissible expression with small equation and constraint residuals.Recovered expressions must be differentiable to the order required by the operators and evaluable without retaining a neural-network representation.Source: §2.2 · pp. 7–8

2.3. Physical consistency and verification measures

  • Normalized governing-equation and constraint residuals measure physical consistency, while coefficient refinement minimizes an unnormalized weighted sum of squared pointwise residuals.The refinement objective is not interpreted as an empirical estimate of the normalized verification measures.Source: §2.3 · p. 8
  • Known reference values are used only for post hoc relative L2 error computation, not as PINN supervision or for checkpoint selection, search, candidate selection, or refinement.Source: §2.3 · p. 9

3. The DeSyR framework

DeSyR separates teacher-guided symbolic topology search from physics-based coefficient refinement, followed by gated selection and verification. The framework also extends this approach to coupled systems, where governing equations jointly determine coefficients.

3. The DeSyR framework

  • DeSyR separates teacher-guided topology proposal and provisional fitting from physics-only coefficient refinement on frozen topologies, then selects candidates through explicit gates.Teacher samples guide search and assess compatibility after refinement, but do not enter the coefficient-refinement objective or verification residuals; the same division applies to coupled multi-field problems.Source: §3 · p. 9

3.1. The search–refinement decoupling principle

  • The search–refinement principle uses teacher samples alone for topology search, then re-estimates constants on fixed trees without teacher data and selects through explicit gates.The exact-recovery pathway operationally depends on adequate teacher guidance, candidate coverage, and refinement convergence; these are not sufficient conditions for exact recovery, and nonlinear refinement may depend on initialization through its basin of attraction.Source: §3.1 · pp. 11–12

3.2. Stage A: teacher-guided structure search

  • Stage A selects a PINN teacher by physics-validation score, then uses its samples for a teacher-fit–complexity Pareto search and pools candidates from independent runs.The search objective contains no governing-equation or constraint residual; retained provisional coefficients only initialize physics refinement.Source: §3.2 · pp. 14–15

3.3. Stage B: physics-based coefficient refinement

  • Stage B freezes each candidate tree and optimizes eligible numerical constants against governing-equation and prescribed-constraint residuals, using multiple initializations for the potentially nonconvex problem.The refinement objective excludes teacher and reference-solution terms; exact coefficient recovery also requires the target coefficients to be representable in the fixed-topology parameterization.Source: §3.3 · pp. 15–17

3.4. Stage C: gated selection and verification

  • Stage C cleans candidates without refitting, then applies sequential convergence, teacher-compatibility, physics-equivalence, and complexity gates before reporting independent verification quantities.If any eligible candidate converges, selection is restricted to converged candidates; within the physics-equivalent class, the least-complex candidate is preferred.Source: §3.4 · pp. 17, 19, 20
  • Interior residuals verify the domain out of sample, while constraint residuals check prescribed boundary or initial data; reference values are reserved for post hoc error reporting.For manufactured benchmarks, relative L2 error uses the unique classical solution as the reference; its pointwise values are otherwise excluded from recovery.Source: §3.4 · p. 20
  • Stage C selects among refined candidates using ordered eligibility, teacher-compatibility, physics-equivalence, and complexity gates, then reports verification quantities.The workflow records convergence status, residuals, and failure reasons for candidates before selecting the final expression.Source: §3.4 · p. 21

3.5. Extension to coupled multi-field systems

  • For coupled systems, fieldwise teacher-guided searches form candidate groups, which are screened for joint physical consistency, mean teacher error, and total complexity before bounded joint refinement.The preliminary screen ranks generated expressions using provisional coefficients and does not change their topologies.Source: §3.5 · p. 22
  • Coupled coefficients are estimated using a joint physics-only objective during screening and full-set refinement, with shared parameters for eligible recurring exponential rates.The joint parameterization represents rates detected in multiple fields as integer multiples of a shared base rate or as field-specific constants; exact recovery requires representability within that parameterization.Source: §3.5 · p. 22
  • For coupled candidates, selection gates operate on jointly screened groups; after a topology group wins, its topologies remain fixed during full-set refinement and the gates are not reapplied.The group-level gates assess screening convergence, teacher compatibility, physics equivalence, and complexity preference.Source: §3.5 · p. 23

4. Theoretical analysis

The analysis separates fixed-topology coefficient refinement from topology discovery: under stated conditions, physics-only refinement can recover exact coefficients, while teacher-driven fitting and finite-weight mixtures can retain teacher error.

4.1. Problem setting and assumptions

  • The analysis treats coefficient refinement after topology is fixed; topology discovery remains outside its guarantees, and guarantees for nonlinear coefficient optimization are local.The exact-recovery result relies on well-posedness, representability, zero-residual attainment, and discrete determinacy.Source: §4.1 · p. 24, §4 · p. 24

4.2. Error inheritance from the teacher

  • In representable linear models, teacher-fit coefficient error reflects the projection of teacher error onto model space, is amplified by 1/σmin(Φ), and need not vanish as sample size grows.If the normalized projected teacher error converges to a nonzero limit, coefficient bias persists; when the topology is not representable, refinement cannot remove its representation-error component.Source: §4.2 · pp. 26–28

4.3. Conditional exact recovery

  • With a fixed topology, well-posedness, representability, zero-residual attainment, and discrete determinacy imply physics-only refinement recovers the unique classical solution exactly.The target is independent of the teacher, but reaching it numerically can depend on the teacher-generated initialization; neighborhood injectivity is a sufficient, not necessary, determinacy condition.Source: §4.3 · pp. 28–29

4.4. Finite-weight bias in mixed objectives

  • For linear fixed-topology models, every finite physics weight retains teacher-dependent coefficient bias when teacher error projects onto the model space; the bias is O(β^-1) as β grows.The bias vanishes when the teacher error is orthogonal to the sampled model space; the result does not establish monotonic error reduction as β increases.Source: §4.4 · pp. 30–31

4.5. Nonlinear local identifiability and convergence

  • For nonlinear parameterizations, a full-rank local Jacobian yields local identifiability, and ideal full-step Newton and Gauss–Newton converge quadratically from sufficiently close initializations.Outside the local convergence neighborhood, convergence is not guaranteed and nonzero-residual spurious stationary points may occur; the teacher affects refinement through initialization.Source: §4.5 · pp. 31–33

5. Numerical experiments

The experiments assess DeSyR across heterogeneous differential-equation problems using controlled comparisons that separate symbolic search from fixed-topology coefficient refinement, with five PINN teachers per configuration and separately seeded searches expanding each teacher’s candidate pool. They examine when refinement and selection yield accurate recoveries, how verification reveals representation errors and missed candidates, and how convergence eligibility and problem-dependent runtime affect the results.

5.2. End-to-end recovery across the benchmark suite

  • Across 18 configurations from 15 problems, all 90 selected expressions converged, and median refined relative L2 errors were 2.31 × 10^-14 or lower.Seven configurations had stored-zero median refined error; Helmholtz had the largest remaining median error, while its median teacher error was 4.94 × 10^-2.Source: Table 4 · p. 36, §5.2 · pp. 36–37

5.3. Representative mechanics cases

  • For Burgers, fixed-topology refinement reduced median relative L2 error from 3.92 × 10^-4 to 3.71 × 10^-17; Helmholtz reached 2.31 × 10^-14.The selected candidates captured the travelling-wave and high-frequency product structures, respectively.Source: §5.3 · pp. 40–41, Figure 4 · p. 41
  • For the coupled Kovasznay flow [28, 23], joint refinement yielded 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 reconstruction captured cross-field structure while enforcing incompressibility.Source: §5.3 · p. 42, Figure 5 · p. 42

5.4.1. Candidate-pool coverage under repeated searches

  • Pooling a second symbolic search increased recovery from 58 of 60 to all 60 teacher-level instances; no additional instances changed for K = 3, 5, or 10.The replay is an end-to-end candidate-coverage indicator, not a direct measurement of algebraic topology coverage, and its saturation point need not generalize to other budgets or distributions.Source: §5.4.1 · p. 43, Figure 6 · p. 44

5.4.2. Fixed-topology coefficient correction

  • On Sine–Poisson 1D, refinement moved α from 3.1415536 to 3.14159265358979 and reduced relative L2 error from 2.48 × 10^-5 to 1.98 × 10^-15.Across representative fixed topologies, refinement also reduced error from 3.90×10^-3 to 7.84 × 10^-17 for Multifrequency Poisson, from 1.51 × 10^-2 to 3.01 × 10^-14 for Euler–Bernoulli, and from 3.92 × 10^-4 to 3.71 × 10^-17 for Burgers, without reopening topology search.Source: §5.4.2 · pp. 44–46, Table 5 · p. 46

5.4.3. Ablation of coefficient-refinement objectives

  • For six linear fixed bases, mixed-objective error tails approached the physics-only solution with slopes close to −1 as β increased, consistent with O(β^-1) finite-weight bias.Across 50 recovered-topology problem–seed combinations, median relative L2 error fell from 8.85 × 10^-5 with teacher-only fitting to 1.45 × 10^-13 at β = 10^6 and 1.86 × 10^-17 with physics-only refinement; no asymptotic rate is assigned to nonlinear cases.Source: §5.4.3 · pp. 47–48, Figure 8 · p. 47, Figure 9 · p. 48

5.4.4. Initialization sensitivity and local identifiability

  • Teacher-derived initialization was more reliable across heterogeneous cases, while all 13 audited scalar free-coefficient Jacobians were numerically full column rank at refined solutions.Random starts sometimes achieved competitive results, but the local rank and conditioning diagnostics do not establish global uniqueness or convergence from arbitrary initializations.Source: §5.4.4 · pp. 49–50, Figure 10 · p. 51

5.4.5. Scope and Representational Limitations of the Operator Library

  • With a polynomial-only operator library, the exact sine solution lies outside the admissible expression space, and refinement yields a sixth-degree polynomial.Source: §5.4.5 · p. 52
  • With a misspecified polynomial library, refinement reached constraint residual Rcon = 6.99 × 10−15, but independent equation residual Req = 6.31 × 10−2 and relative L2 error 5.36 × 10−4 remained.Verification on independent interior points exposes representation error that coefficient refinement cannot eliminate when the target is absent from the prescribed search library.Source: §5.4.5 · p. 52, Figure 11 · p. 53
  • In the frozen-teacher study, enrichment reduced Helmholtz recovery from five of five teachers to two of five, while K = 5 restored five of five at higher median time.For Burgers, enrichment changed recovery from 4/5 to 3/5 and median time from 275.5 to 377.8 s; for Sine–Poisson 1D, all five teachers recovered the same expression in both settings. Failed enriched Helmholtz searches had no target-family candidate in either raw Pareto front or the retained pool; the results do not establish a generally sufficient K.Source: §5.4.5 · p. 53, Table 6 · p. 54

5.5.1. Stage-C selection-gate ablation

  • Removing the convergence-eligibility gate raised worst relative L2 error from 1.16×10−13 to 4.29×10−3 in 60 frozen candidate pools.The replacement was an unconverged complexity-47 candidate; removing complexity preference changed 12 selections without increasing worst error, while two other gates were inactive in these pools.Source: §5.5.1 · p. 54
  • 3,745 of 3,774 candidate refits (99.23%) converged across 17 archived scalar configurations; the remaining 29 were timeouts or finite unconverged refits.No non-finite outcomes were observed; these statistics support the operational convergence check but do not establish a global convergence guarantee.Source: §5.5.1 · pp. 54–55

5.5.2. Wall-clock cost

  • Median total runtimes ranged from approximately 100 seconds for Parametric Poisson to roughly 18–25 minutes in the most computationally demanding configurations.PINN training dominated fourth-order Euler–Bernoulli, whereas symbolic recovery dominated Wave and Burgers; timings are implementation- and hardware-dependent empirical cost indicators, not asymptotic estimates.Source: §5.5.2 · p. 55

6. Discussion

DeSyR separates teacher-guided topology search from physics-only coefficient refinement, while retaining teacher compatibility as a selection gate. Its guarantees depend on candidate coverage and refinement conditions, and its discussion identifies limitations and directions for broader evaluation.

6.1. Methodological implications and diagnostics

  • Teacher data guide topology search and compatibility screening, but the governing equation and constraints alone determine reported coefficients after the topology is fixed.This separates DeSyR from teacher-only distillation and mixed data–physics coefficient objectives.Source: §6.1 · p. 55
  • For linear fixed-topology models, a finite-weight mixed objective retains an O(β^-1) teacher-dependent contribution when Φ^Tε ≠ 0; physics-only refinement conditionally removes this bias source.The conditions are well-posedness, fixed-topology representability, zero-residual attainment, and discrete determinacy.Source: §6.1 · pp. 55–56
  • Recovery requires informative teacher-based ranking and compatibility, a retained target-capable topology, and at least one such candidate to yield finite, converged refinement.These conditions also make recovery dependent on the prescribed expression library.Source: §6.1 · p. 56
  • 6.31×10−2 was the independently evaluated equation residual for a polynomial-library expression that satisfied prescribed constraints, exposing a representation-space mismatch.Initialization and Jacobian audits support teacher-derived starting values and local rank/conditioning assessment; frozen-pool gate replay shows convergence eligibility can avert severe selection failure. Enrichment can require larger search budgets to retain target-capable topologies; the diagnostics do not establish library-independent coverage or global convergence.Source: §6.1 · p. 56
  • Exact fixed-topology recovery on finite collocation sets requires sufficient discrete determinacy; nonlinear identifiability and convergence guarantees are local and depend on Jacobian conditions.These results do not guarantee topology discovery or global nonlinear convergence.Source: §6.1 · p. 56

6.2. Scope and future directions

  • The study uses controlled benchmarks with prescribed operator libraries, regular domains, and analytic references; extensions to noisy data, irregular geometries, discontinuities, multiscale solutions, or richer symbolic spaces need renewed evaluation.Adaptive operator screening or hierarchical library expansion guided by teacher information and physics is proposed to improve coverage while controlling combinatorial growth.Source: §6.2 · p. 57
  • All 48 fixed-topology refits converged under constraint perturbations, but error relative to the unperturbed reference rose with perturbation level.This is limited evidence about Stage-B sensitivity, not a general stability guarantee or evidence about noise in teacher-guided topology discovery.Source: §6.2 · p. 57
  • Future work may test recovered expressions in sensitivity analysis, reduced-order modeling, parameter studies, and repeated design-optimization evaluation.Explicit symbolic forms may offer interpretability and computational-efficiency advantages in these downstream uses.Source: §6.2 · p. 57

7. Conclusions

DeSyR separates PINN-guided symbolic topology discovery from coefficient refinement driven only by the governing equation and prescribed constraints. Across tested problems and configurations, refinement often reaches near-machine-precision accuracy, subject to topology representability and convergence.

7. Conclusions

  • A PINN guides stochastic symbolic structure search and supplies provisional coefficients, while physics and prescribed constraints alone define refinement after the topology is fixed.An approximate teacher need not impose its error scale on final coefficients if a target-capable topology is retained and physics-only refinement converges.Source: §7 · pp. 57–58
  • For linear fixed-topology models, physics-only refinement conditionally recovers exact coefficients; nonlinear identifiability and convergence guarantees are local and make initialization explicit.The analysis characterizes teacher-error inheritance and finite-weight bias from mixed data–physics objectives; exact recovery assumes well-posedness, representability, zero-residual attainment, and discrete determinacy.Source: §7 · pp. 57–58
  • Across 15 differential-equation problems and 18 configurations, refinement reaches near-machine-precision accuracy in many cases and cuts positive same-topology relative L2 errors by eight to fourteen orders of magnitude.Nonlinear and coupled cases show the procedure can operate beyond linear scalar problems.Source: §7 · p. 58
  • Coefficient refinement cannot remove representation error when the target lies outside the symbolic space, and enlarging the operator library can reduce candidate coverage under a finite search budget.Retaining a target-capable topology may therefore require additional search effort.Source: §7 · p. 58

Appendix A. Proofs of theoretical results

The proofs establish conditional exact recovery for fixed topologies and local guarantees for nonlinear refinement. They also show that teacher data affect physics-only refinement through its initialization, not its fixed-topology objective.

Appendix A. Proofs of theoretical results

  • For full-column-rank Φ, the linear least-squares fit satisfies the normal equations, and its data-to-coefficient operator norm is 1/σmin(Φ).The proof obtains this sensitivity from the thin SVD of Φ.Source: Appendix A · p. 59
  • Under Assumptions 4.1–4.4, zero residual and local injectivity of F imply that the refined coefficients equal a⋆ and the recovered solution equals u⋆.The classical solution makes F(a⋆) = 0; if F is injective on a neighborhood containing both a⋆ and the refit, the zero-residual refit must be a⋆.Source: Appendix A · p. 59
  • For fixed topology, coefficient parameterization, and refinement points, the refinement objective uses only the governing equation and prescribed constraints; teacher dependence enters only through initialization a0(θ).The proof states that the objective contains no teacher samples or reference-solution values once those fixed objects are specified.Source: Appendix A · p. 59
  • For every β > 0, the mixed-fit objective is strictly convex; as β → 0+, its solution tends to the teacher least-squares solution.The proof establishes positive definiteness of the Hessian from the positive-definite data and physics matrices.Source: Appendix A · pp. 59–60
  • In nonlinear refinement, the stated rank and regularity conditions yield local strong convexity, a unique zero-residual point near a⋆, and quadratic local Newton or Gauss–Newton convergence.At the zero-residual solution, the Gauss–Newton Hessian approximation coincides with the exact Hessian.Source: Appendix A · p. 60

Appendix B. Benchmark definitions and fixed configurations

Appendix B defines the benchmark problems and documents the fixed numerical and search-operator settings used across 18 configurations.

Appendix B. Benchmark definitions and fixed configurations

  • Table B.1 summarizes the 18 configurations’ governing equations, computational domains, prescribed constraints, reference solutions, and problem-specific unary operators.It also defines the analytic reference solution notation and spatial-boundary notation used for time-dependent problems.Source: Appendix B · p. 61, Table B.1 · p. 62
  • Complete search-operator settings are provided in Table B.3.Source: Appendix B · p. 62
  • Fixed numerical settings are consolidated by configuration ID in Tables B.2 and B.3.Source: Appendix B · p. 63

Appendix B.1. PINN and sampling settings

Table B.2 documents the PINN architectures and sampling conventions used across 18 configurations, including how training and validation/refinement point counts are interpreted.

Appendix B.1. PINN and sampling settings

  • Table B.2 covers PINN architectures and sampling settings for 18 configurations, with 3 × 40 denoting three hidden layers of 40 neurons each.The table defines the architecture notation used for its configuration entries.Source: Table B.2 · p. 63
  • Training-point counts are ordered interior/boundary/initial, and a dash marks an unused class; counts denote sampled locations, not scalar residual components.The Kovasznay pressure anchor is an additional point constraint excluded from the boundary-location count.Source: Table B.2 · p. 63
  • NS counts frozen-teacher samples for Stage-A symbolic search, while N_val and N_r count interior equation-residual points for physics validation and Stage-B refinement, respectively.Constraint terms use their separately prescribed constraint point sets.Source: Table B.2 · p. 63, Appendix B.1 · p. 63

Appendix B.2. Symbolic-search and refinement settings

Appendix B.2 specifies the symbolic-search and refinement settings across 18 configurations, including additional coupled-search procedures for Kovasznay.

Appendix B.2. Symbolic-search and refinement settings

  • For each PINN teacher, 10 independently seeded searches retain up to five representative candidates under the Stage-A Pareto-front rule.Configuration-specific refinement-point counts are reported in Table B.2.Source: Table B.3 · p. 64
  • For coupled Kovasznay, the top 16 candidate groups undergo joint screening refinement before the selected topology group is re-estimated on all 6000 refinement points.Groups are formed from fieldwise shortlists of at most 12 candidates per field, evaluated on 750 points, and ranked by joint physics score, mean teacher-relative L2 error, then search complexity; screening uses one initialization and at most 1000 residual evaluations.Source: Appendix B.2 · p. 64
  • Shared-factor augmentation scans exp(rxq) factors with a single-variable linear exponent and |r| ≥ 10^-10, retaining up to two eligible rate clusters and at most three trigonometric factors per field.Clusters require support from at least two fields; retained rates and factors generate at most six additional candidates per field, selected by alternating teacher-fit and complexity rankings.Source: Appendix B.2 · pp. 64–65
  • For Kovasznay, rates matched across at least two fields define shared base parameters; rates matching a base-rate multiple k ∈ {1, 2, 3, 4} within 10^-8 are tied to it, while others remain field-specific.Matching tests the ordered base-rate list and uses the first qualifying base rate.Source: Appendix B.2 · p. 65

Appendix C. Detailed numerical results

Appendix C supplements the configuration-level medians reported in the main text with across-seed performance distributions for 18 benchmark configurations.

Appendix C. Detailed numerical results

  • The consolidated table reports across-seed performance distributions for 18 benchmark configurations.It avoids repeating the same configurations in multiple class-specific layouts.Source: Appendix C · p. 65, Table C.1 · p. 66
  • Relative L2 errors are summarized as median [first quartile, third quartile] over five independently initialized PINN teachers.Source: Table C.1 · p. 66
  • Pre-refit identifies the search-stage expression corresponding to the candidate ultimately selected after refinement and Stage-C selection.The complexity column gives the range of final expression complexity across the five seeds.Source: Table C.1 · p. 66

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

The appendix audits candidate coverage in a Helmholtz library-enrichment study and reports how repeated searches contribute to first-K end-to-end recovery analysis. It distinguishes candidate-pool expansion for a fixed teacher from independent teacher replicates.

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

  • Table C.2 reports first-K replay recoveries among five independently trained PINN teachers, with success defined as selected refined-expression relative L2 error no larger than 10^-10.Its last column gives the smallest tested K that recovers all five teacher-level instances.Source: Appendix C.1 · p. 67, Table C.2 · p. 67
  • Separately seeded symbolic searches enlarge the candidate pool for a fixed teacher; they are not additional independent replicates.Source: Appendix C.1 · p. 67
  • Across all three Helmholtz library-enrichment settings, raw Pareto-front and retained Stage-B counts coincide, so no audited target-family candidate is lost during retention.The audit looks for candidates in the prescribed family C sin(ax + b) sin(cy + d) + e.Source: Appendix C.1 · p. 67, Table C.3 · p. 68
  • Because the Helmholtz audit uses a strict structural predicate, it may miss expression trees equivalent to the target only after coefficient refinement and symbolic simplification.Source: Appendix C.1 · p. 67

Appendix C.2. Coefficient-refinement objective ablation · Appendix C.3. Initialization sensitivity

Appendix C.2 reports fixed-topology objective-ablation tests, while C.3 examines initialization sensitivity. The passages specify experiment designs and some tail slopes, but do not provide the full outcome tables.

Appendix C.2. Coefficient-refinement objective ablation

  • Each row uses five independently initialized PINN teachers, for 30 teacher instances in total.Source: Appendix C.2 · p. 69
  • The theory-aligned experiment reports fixed bases and slope estimates for linear parameterizations that exactly represent the solution.The tail slope is fit to log10 relative L2 error versus log10 β over the last three finite-weight points whose median error exceeds the numerical-accuracy floor.Source: Table C.4 · p. 69
  • Tail slopes were −1.000 for Multifrequency Poisson and −0.999 for Euler–Bernoulli in the theory-aligned fixed-topology experiment.Each basis admits an exact solution representation and is linear in its coefficients.Source: Appendix C.2 · p. 69, Table C.4 · p. 69
  • The objective ablation covers 50 problem–seed combinations from ten scalar configurations with free coefficients.Table C.5 reports relative L2 errors and equation residuals as median [first quartile, third quartile]; residuals use independent interior verification points.Source: Table C.5 · p. 69

Appendix C.3. Initialization sensitivity

  • For each of five PINN seeds per configuration, C.3 compares one teacher-derived initialization with five random initializations drawn from U(−1, 1).This gives nT = 5 and nR = 25 per configuration across ten scalar configurations with free coefficients.Source: Table C.6 · p. 70
  • C.3 reports post-refinement relative L2 errors as median [first quartile, third quartile].Source: Table C.6 · p. 70

Appendix C.4. Sensitivity to Perturbations in Prescribed Constraints · Appendix C.5. Rank and conditioning diagnostics · Appendix D. Additional numerical figures

The appendices test how prescribed-constraint noise affects fixed-topology refinement, report rank and conditioning diagnostics, and provide additional recovery, seed-level, timing, and selection-gate visualizations.

Appendix C.4. Sensitivity to Perturbations in Prescribed Constraints

  • Constraint sensitivity was tested by adding independent Gaussian noise only to prescribed boundary or initial values while keeping the equation unperturbed and topology target-capable.For each of three problems, five draws were used at each nonzero η ∈{10−6, 10−4, 10−2}; errors were measured against the unperturbed reference solution and residuals at independent interior points.Source: Appendix C.4 · p. 71
  • 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.Across all 48 refits, including three clean baselines, every run converged; solution error and governing-equation residual generally increased with η, with problem-dependent sensitivity. The evidence is limited to the tested fixed-topology protocol and does not establish general noise stability or end-to-end discovery under noisy observations.Source: Appendix C.4 · p. 71, Figure C.1 · p. 72

Appendix C.5. Rank and conditioning diagnostics

  • Table C.7 reports discrete refinement-residual Jacobian rank and conditioning diagnostics at refined coefficients, with free-coefficient metrics summarized across five PINN seeds.Full-rank counts come from a finite-precision rank audit; zero-parameter runs are reported separately because rank and conditioning metrics do not apply.Source: Table C.7 · p. 73

Appendix D. Additional numerical figures

  • Figures D.1–D.5 show representative one-dimensional, space–time, two-dimensional, and three-dimensional recovery profiles or fields alongside PINN and refined-expression errors.Representative profiles and fields use the seed whose PINN relative L2 error is closest to the five-seed median; methods within a configuration share that seed, and error maps use matched logarithmic scales within each case.Source: Appendix D · p. 74, Figure D.1 · p. 74, Figure D.2 · p. 75
  • Figures D.6 and D.7 report seed-level relative L2 errors and final expression complexity across five PINN seeds for all 18 configurations.The error figure distinguishes PINN, pre-refit, and refined errors; the complexity figure notes that Kovasznay’s spread reflects algebraically equivalent coupled-solution representations.Source: Figure D.6 · p. 78, Figure D.7 · p. 79
  • Figure D.8 and Table D.1 present configuration-level and stage-wise wall-clock timing summaries, with recovery including symbolic search, refinement, selection, and verification.The timing captions note missing or differing run counts for some configurations, checkpoint-loaded teachers in selected totals, and that timings depend on hardware and implementation.Source: Figure D.8 · p. 80, Table D.1 · p. 81
  • In replay on 60 frozen scalar candidate pools, removing the convergence-eligibility gate admitted one unconverged candidate and substantially worsened the worst selected relative L2 error.Removing the complexity preference changed 12 selections without increasing worst error; teacher-compatibility and physics-equivalence gates were inactive in these pools.Source: Figure D.9 · p. 82

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