Selective boundary condition reduction via learned error gating

1. Introduction

• The finite-parameter suitability of a limiting Dirichlet law depends on forcing, coefficients, geometry, and error norm; domain accuracy alone can hide a larger boundary error relevant to coupling.Classical convergence analyses establish the limit, but do not answer this instance-specific decision.Source: §1 · p. 1

1.1. Contributions.

• Paired full and limit solutions train a selector to predict domain and boundary errors, so a solver uses the simpler law only when both meet user tolerances.The full law remains the fallback, tolerances can change without retraining, and split conformal calibration can make both estimates conservative.Source: §1 · p. 1, §1.1 · p. 2
• Three benchmarks cover harmonic galvanic corrosion, stationary nonlinear transfer, and nonlinear evolution; the stationary benchmark reports about 133× speedup for limit-model selection and 8.3× for the complete policy.Complete-policy timings include feature construction, inference, selected solves, and full fallbacks; offline preparation is excluded.Source: §1.1 · p. 2

1.2. Related work.

• Prior Robin-to-Dirichlet analyses establish or discretize the limit, while neural surrogates can be unreliable near singular phenomena; this selector instead learns decision errors.Its specialization keeps the state space, interior operator, and discretization fixed while changing only the boundary law; split conformal calibration gives marginal coverage under exchangeability.Source: §1.2 · p. 2

2.1. A preliminary elliptic example.

• As κ decreases, the Robin boundary law enforces the target Dirichlet value, but a nonzero target jump can make the admissible Dirichlet class empty and require a weaker trace interpretation.The jump case is the setting of Experiment 1.Source: §2.1 · pp. 2–3

2.2. Full and limit models.

• The full and limit models share the interior operator, source, geometry, and conditions away from Γ; only the law on Γ changes, from a possibly nonlinear response to u = uD(x).Smaller κ corresponds to a stiffer boundary response.Source: §2.2 · p. 3
• The stated limit assumptions require well-posed models, a locally unique uniformly strongly monotone boundary response, and convergence in the domain and boundary norms used for selection.At jumps in uD, convergence may instead be in weaker trace norms and locally in the interior.Source: §2.2 · p. 3

2.3. Error measures and the reference rule.

• The reference rule accepts the limit model only when both relative errors, EΩ and EΓ, meet their tolerances; the practical rule estimates both before the full solve.For an evolutionary trajectory, the errors are maximum discrepancies over stored time levels, and the benchmark accepts only when both predicted maxima meet tolerance.Source: §2.3 · p. 4

3. Method for selecting boundary laws

• Offline paired solves train domain and boundary error estimates; online, the limit law is selected only when both predicted errors meet their tolerances.The selector returns the solution from the chosen model.Source: §3 · p. 4, Paired solutions and errors to be… · p. 5, Figure 2 · p. 5

Choosing the error estimate and applying the rule

• Predicting two scalar errors keeps output size independent of the mesh, permits tolerance changes without retraining, and leaves field computation to a PDE solver.Inputs derived from the limit solution require solving that model before selection, so savings depend on avoided full-solve cost exceeding this overhead.Source: Choosing the error estimate and… · pp. 4–5

Paired solutions and errors to be estimated

• Training cases pair full and limit solutions, recording stationary errors or evolutionary trajectory errors for an estimator that outputs log10 error estimates.The estimator predicts separate domain and boundary discrepancies.Source: Paired solutions and errors to be… · p. 5
• Optional split conformal calibration scales both predicted errors so that, under exchangeability, both true errors are no larger with probability at least 1 − α.This bounds marginal unsafe-selection probability, not the unsafe fraction among accepted cases, and does not protect against distribution shift.Source: Paired solutions and errors to be… · pp. 5–6

4. Numerical results

• Independent cases from three benchmarks were evaluated; online totals include feature construction, inference, selected solves, and full-model fallbacks, while paired labeling and training are offline.Source: §4 · p. 6

4.1. Experiment 1: harmonic galvanic corrosion model with a boundary jump.

• The corrosion benchmark varies κ, equilibrium potentials, and current scales, measuring errors in the domain and on the active boundary.Source: §4.1 · p. 6
• At 5% domain and boundary tolerances, across repetitions, calibration reduced conditional unsafe rates from 0.87% to 3.48% for the raw selector to 0 to 0.93%, withdrawing four acceptances, including one unsafe choice, and increasing missed safe cases from two to five.Source: §4.1 · pp. 6–7

4.2. Experiment 2: stationary nonlinear transfer.

• At nominal tolerances, accepted limit paths accelerate the stationary benchmark, though fallbacks reduce but do not eliminate the policy gain; its finite-test descriptive 95% upper endpoint for unsafe use is 6.2%, not a guarantee.Source: §4.2 · p. 8, Table 2 · p. 8

4.3. Experiment 3: evolutionary nonlinear transfer.

• At nominal tolerances, the larger evolutionary test set had one unsafe accepted case and no missed safe cases; a separate direct policy evaluation had one borderline boundary miss, so both results are descriptive rather than safety guarantees.Accepted limit paths were much faster than full solves, but fallbacks substantially reduced policy speedup.Source: §4.3 · p. 8

4.4. Tolerance versus speedup.

• Relaxing tolerances increased limit use and policy speedup in both benchmarks; Figure 4 marks observed unsafe selections with open rings.At λ = 1, the stationary sweep agrees with Table 2, while the evolutionary result uses the larger test set rather than the separate policy evaluation.Source: §4.4 · p. 9, Figure 4 · p. 9

• Tests on stationary and evolutionary PDEs show that tolerance, limit use, reusable operators, and fallbacks determine policy speedup.Source: §5 · p. 9
• The selector predicts domain and boundary errors, lets a PDE solver enforce the chosen law, retains the full law as fallback, and permits tolerance changes without retraining.Source: §5 · p. 9
• Calibration bounds the marginal probability of an unsafe choice, but not tolerance violations conditional on selecting the limit model; conditional-risk control and calibration sample sizing remain open.Source: §5 · p. 9
• Evolutionary tests select one model for a whole trajectory and check stored time levels; between-level control needs temporal error estimates, while adaptive switching must track accumulated error.Source: §5 · p. 9
• For boundary laws beyond linearization, proposed prediction features include justified convergence rates, boundary layers, and junction corrections, while accounting for nonsmooth-data changes in rates or norms.Source: §5 · p. 10

Appendix A. Technical implementation details

• Calibration sorts normalized errors on training-unused cases and sets c to at least 1 from the kth score, or infinity when k exceeds ncal.The score is the maximum across outputs of each error divided by its bound.Source: Common protocol · p. 11

Common protocol.

• Calibration and test cases are independent samples from the design, and test sets are disjoint.The design ranges are log10 κ ∈ [−7, −3], ϕa ∈ [−0.26, −0.14], ϕc ∈ [0.14, 0.26], ic,0 ∈ [1.5, 6] × 10−4, and ia,0 ∈ [1.5, 6] × 10−2; currents are sampled logarithmically.Source: Experiment 1: Corrosion · p. 11, Table A.1 · p. 11

Experiment 1: Corrosion.

• The inputs also include the five parameters, the potential jump ϕc−ϕa, and two linearized indicators.The limit flux qh supplies the correction δΓ = −κqh/i′∗(ϕ∗).Source: Experiment 1: Corrosion · p. 12
• The correction and its harmonic extension give bΩ and bΓ, and the targets are mj − log10 bj.Source: Experiment 1: Corrosion · p. 12
• Inputs include log10 κ, log10(1 + γ), five other values, and log10(κL), with L defined from f1, f2, gx, and gy.Samples use log10 κ ∈ [−5, −0.5], log10 γ ∈ [4, 8], g0 ∈ [0.8, 1.2], gx ∈ [−0.3, 0.3], gy ∈ [−0.25, 0.25], f1 ∈ [0, 8], and f2 ∈ [−4, 4].Source: Experiment 2: Stationary · p. 12

Experiment 2: Stationary.

• Evolutionary samples vary log10 κ, ν, β, ag, ωg, af, and ψ across the stated ranges.The ranges are log10 κ ∈ [−3.5, −0.5], ν ∈ [0.035, 0.14], β ∈ [20, 120], ag ∈ [0.06, 0.30], ωg ∈ [0.5, 2.3], af ∈ [0.15, 1.10], and ψ ∈ [0, 2π].Source: Experiment 2: Stationary · p. 12
• Inputs combine scaled parameters with log10 κ/ν, βag, and sine and cosine features of ψ.Labels use backward Euler, while checks use centered differences of order two.Source: Experiment 2: Stationary · p. 12
• Full solves use damped Newton, and selected trajectories reuse the limit factorization.Source: Experiment 2: Stationary · p. 12

Timing and refinement.

• Table 2 timings use 24 cases and seven repeats; policy speedups use 64 tests, three repeats, and six fallbacks.Single-thread timings exclude training and the initial factorization.Source: Timing and refinement · p. 12
• Sweeps time each path once, including prediction and fallback.Source: Timing and refinement · p. 12
• Refining one case per study to 412, 1292, and 1292 steps changed errors by at most 2.4 × 10−3, 1.8 × 10−7, and 2.2 × 10−4, respectively, with no decision changes.The last refinement used 256 steps.Source: Timing and refinement · p. 12