A Greedy PDE Router for Blending Neural Operators and Classical Methods

Sahana Rayan; Yash Patel; Ambuj Tewari

Hyperparameter	Paired study	Growing ensembles
Feature variants	Scalar	Scalar; pooled residual
Hidden width	$64$	$64$ scalar; $32$ pooled
Instances per round	$128$	$256$
DAgger rounds	$2$	$4$
Epochs per fit	$150$	$150$
Optimizer	Adam	Adam

Appendix

Table 3: Router hyperparameters. A DAgger round denotes an additional router-policy rollout and oracle relabeling pass after the initial exploratory oracle round.

Equation	Action	Primitive cost (ms)	$m$	Macro cost (ms)
Poisson	WJacobi(.67)	0.147	6	0.884
	GS	0.443	2	0.886
	SOR(1.5)	0.448	2	0.896
	MG	3.614	1	3.614
	Global $\mathrm{s32}$	0.897	1	0.897
	Local $3\times 3$	0.370	1	0.370

Appendix

Table 4: Measured complete-action costs for the

N=128

experiments. Costs include the update and subsequent residual refresh. For growing-ensemble actions,

m

is the number of primitive applications in one macro-action and the last column is

mc_{j}

. The

\mathrm{s32}

cost is the matching reference. The full global model is used only by paired controls, so no growing-ensemble macro count is assigned to it.

Equation	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$
Poisson	WJacobi(.67)	Local $3\times 3$	Global $\mathrm{s32}$	GS	SOR(1.5)	MG
ConvDiff	WJacobi(.67)	Local $3\times 3$	Global $\mathrm{s32}$	GS	SOR(1.5)	MG
AnisoDiff	WJacobi(.67)	Local $3\times 3$	GS	SOR(1.5)	Global $\mathrm{s32}$	MG

Appendix

Table 5: Frozen action order for the

N=128

growing-ensemble experiments. Each size-

K

action set is the length-

K

prefix of the corresponding row.

Classical solver	Poisson	ConvDiff	AnisoDiff
Jacobi	$50$ ; 785.88 (92.58)	$50$ ; 1190.68 (125.93)	$50$ ; 353.62 (60.96)
WJacobi(.67)	$10$ ; 17.34 (0.36)	$10$ ; 18.23 (0.41)	$25$ ; 23.16 (0.15)
GS	$5$ ; 28.88 (0.38)	$5$ ; 30.93 (0.37)	$25$ ; 69.99 (1.43)
SymGS	$5$ ; 30.60 (0.03)	$5$ ; 31.64 (0.21)	$5$ ; 29.02 (0.59)
SOR(1.5)	$10$ ; 36.74 (0.50)	$10$ ; 43.53 (0.58)	$10$ ; 42.50 (0.62)
MG	$5$ ; 23.44 (0.21)	$5$ ; 26.37 (0.24)	$5$ ; 57.12 (2.06)

Appendix

Table 6: Post-selected HINTS results at relative solution error

10^{-8}

on the

64

shared

N=128

test cases. Each cell reports the selected period

\tau

and mean selected-action replay time in milliseconds (standard error in parentheses). The period is the best of

\{5,10,25,50\}

on these same cases, so these are optimistic HINTS results rather than held-out cadence estimates.

PDE	WJ(.67)	GS	SOR(1.5)	MG	Global neural	Local convolution
Poisson	$0.997581$ (exact)	$1.019231$ (SV)	$1.369303$ (SV)	$\geq 0.036064$ (LB)	$\geq 1$ (exact witness)	$1.064909$ (exact)
ConvDiff	$0.998390$ (exact)	$1.022788$ (SV)	$1.475764$ (SV)	$\geq 0.024636$ (LB)	$\geq 1$ (exact witness)	$1.016401$ (exact)
AnisoDiff	$0.999936$ (exact)	$1.050241$ (SV)	$1.459752$ (SV)	$\geq 0.853678$ (LB)	$\geq 1$ (exact witness)	$35.822662$ (exact)

Appendix

Table 7: Contraction diagnostics for finalized

N=128

macro-actions in the mean-zero Euclidean solution-error norm. “Exact” denotes Fourier diagonalization of the deployed linear map, “SV” a matrix-free largest singular-value estimate, and “LB” a sampled lower bound rather than an upper certificate. Values greater than one falsify strict contraction.

PDE	Undamped Jacobi	SymGS
Poisson	$1.000000$ (44; exact)	$0.985629$ (8; SV)
ConvDiff	$1.000000$ (27; exact)	$0.989307$ (7; SV)
AnisoDiff	$1.000000$ (32; exact)	$1.048533$ (7; SV)

Appendix

Table 8: Additional contraction diagnostics for actions appearing in the finalized pairwise experiments but not in the growing action pool. Parentheses give the frozen number of primitive iterations per routed macro.

PDE	Median $r$	95th percentile	99th percentile	Fraction $r\leq 1.05$
Poisson	$0.4666$	$0.9976$	$1.0649$	$98.96\%$
ConvDiff	$0.4706$	$0.9984$	$1.0081$	$99.38\%$
AnisoDiff	$0.7496$	$11.7255$	$24.4790$	$83.22\%$

Appendix

Table 9: Distributional postfix diagnostics over all ordered one-macro action pairs and the declared

N=128

probe set. Quantiles are for the norm ratio

r

; the slack for the squared-error objective is

\mu=r^{2}

.

	Poisson		ConvDiff		Anisotropic diffusion
Method	Time (ms)	$p$ -value	Time (ms)	$p$ -value	Time (ms)	$p$ -value
Jacobi-related solvers
Jacobi Only	6428.52 (4.15)	$<10^{-3}$	10146.76 (14.21)	$<10^{-3}$	–	–
HINTS-Jacobi	2288.27 (125.46)	$<10^{-3}$	3278.90 (161.52)	$<10^{-3}$	1845.77 (126.04)	$<10^{-3}$
Learned Greedy-Jacobi	1280.80 (64.97)	–	2900.52 (181.73)	–	1117.90 (77.21) [62/64]	–
True-Greedy-Jacobi	1226.02 (66.29)	–	2120.53 (103.75)	–	1130.38 (75.91)	–

Appendix

Table 10: Selected-action wall-clock replay time to relative solution error

10^{-12}

, reported as mean (standard error) in milliseconds over 64 shared test instances. Brackets indicate incomplete coverage. Paired one-sided

t

-tests compare each baseline with Learned Greedy under the alternative

T_{\mathrm{Learned}}<T_{\mathrm{baseline}}

, using jointly successful instances. A dash indicates that no paired test is available. True-Greedy is an oracle and is included as a reference.

Figure 3: Convergence histories for representative test instances. Rows: Jacobi, Gauss-Seidel, Symmetric Gauss-Seidel, Successive Overrelaxation (

\omega=1.5

), and Multigrid. Columns: Poisson, Convection Diffusion, and Anisotropic Diffusion. Greedy yields the fastest monotone decay.

Figure 4: Sample predictions for the Poisson equation across different solver pairings. Each row corresponds to a numerical solver (Jacobi, Gauss–Seidel (GS), Symmetric GS, Damped Jacobi, and Successive Over-relaxation, MG), while columns show predictions from the corresponding solver-only baseline, HINTS, and the learned greedy router. The ground truth solution is shown on the right.

Figure 5: Sample predictions for the convection diffusion equation across different solver pairings. Each row corresponds to a numerical solver (Jacobi, Gauss–Seidel (GS), Symmetric GS, Damped Jacobi, and Successive Over-relaxation, MG), while columns show predictions from the corresponding solver-only baseline, HINTS, and the learned greedy router. The ground truth solution is shown on the right.

Figure 6: Sample predictions for the anisotropic diffusion equation across different solver pairings. Each row corresponds to a numerical solver (Jacobi, Gauss–Seidel (GS), Symmetric GS, Damped Jacobi, and Successive Over-relaxation, MG), while columns show predictions from the corresponding solver-only baseline, HINTS, and the learned greedy router. The ground truth solution is shown on the right.

Figure 7: Pointwise error maps for Poisson sample predictions across solver families. Columns correspond to different numerical solvers, while columns show solver-only, HINTS, and learned greedy router. Errors are computed as

u_{h}-u_{h}^{(T)}=e_{h}^{(T)}

. The learned greedy router consistently yields lower-magnitude errors compared to both solver-only and HINTS baselines.

Figure 8: Pointwise error maps for convection diffusion sample predictions across solver families. Columns correspond to different numerical solvers, while columns show solver-only, HINTS, and learned greedy router. Errors are computed as

u_{h}-u_{h}^{(T)}=e_{h}^{(T)}

. The learned greedy router consistently yields lower-magnitude errors compared to both solver-only and HINTS baselines.

Figure 9: Pointwise error maps for anisotropic diffusion sample predictions across solver families. Columns correspond to different numerical solvers, while columns show solver-only, HINTS, and learned greedy router. Errors are computed as

u_{h}-u_{h}^{(T)}=e_{h}^{(T)}

. The learned greedy router consistently yields lower-magnitude errors compared to both solver-only and HINTS baselines.

Equation	Poisson				ConvDiff				Anisotropic diffusion
Method	$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$	$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$	$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$
Jacobi-related solvers
Jacobi Only	$0.606\;(0.034)$	$<10^{-3}$	$0.697\;(0.034)$	$<10^{-3}$	$0.659\;(0.035)$	$<10^{-3}$	$0.737\;(0.033)$	$<10^{-3}$	$0.649\;(0.031)$	$<10^{-3}$	$0.737\;(0.029)$	$<10^{-3}$
HINTS-Jacobi	$0.005\;(9.55\!\times\!10^{-4})$	$<10^{-3}$	$0.404\;(0.017)$	$<10^{-3}$	$0.005\;(0.001)$	$1.000$	$0.535\;(0.022)$	$<10^{-3}$	$0.013\;(0.003)$	$1.000$	$0.452\;(0.014)$	$<10^{-3}$
Learned Greedy-Jacobi	$0.004\;(7.74\!\times\!10^{-4})$	–	$0.259\;(0.001)$	–	$0.005\;(0.001)$	–	$0.263\;(0.002)$	–	$0.013\;(0.003)$	–	$0.248\;(0.004)$	–
GS-related solvers

Appendix

Table 11: Relative residual at a fixed 25 ms cost-calibrated selected-action budget and time-averaged residual AUC (lower is better). Values are mean (standard error) over 64 shared test instances. We use

R(t)=\|f_{h}-\mathcal{L}_{h}^{a}u_{h}(t)\|_{2}/\|f_{h}\|_{2}

and

\mathrm{AUC}_{25}=(25\,\mathrm{ms})^{-1}\int_{0}^{25\,\mathrm{ms}}R(t)\,dt

. The state is held constant between completed actions; complete update and residual-refresh costs, the initial residual, and learned-router inference are charged. A learned routing decision is admitted only when its complete cost-matched macro-action fits in the remaining budget. Each reported

p

-value is from a paired one-sided

t

-test of the stated baseline against Learned Greedy, with the alternative that Learned Greedy is lower.

Actions	Method	Poisson				ConvDiff				Anisotropic diffusion
		$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$	$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$	$R(25\,\mathrm{ms})$	$p$	$\mathrm{AUC}_{25}$	$p$
$\mathcal{A}_{3}$	Best pairwise router	$1.23\!\times\!10^{-8}\;(1.75\!\times\!10^{-9})$	$<10^{-3}$	$0.057\;(2.07\!\times\!10^{-3})$	$<10^{-3}$	$2.25\!\times\!10^{-6}\;(2.39\!\times\!10^{-7})$	$<10^{-3}$	$0.068\;(2.43\!\times\!10^{-3})$	$<10^{-3}$	$4.18\!\times\!10^{-7}\;(2.84\!\times\!10^{-8})$	$<10^{-3}$	$0.279\;(7.57\!\times\!10^{-3})$	$<10^{-3}$
$\mathcal{A}_{3}$	Router $(\mathcal{A}_{3})$	$1.04\!\times\!10^{-13}\;(6.06\!\times\!10^{-15})$	–	$0.055\;(1.67\!\times\!10^{-3})$	–	$1.13\!\times\!10^{-11}\;(2.30\!\times\!10^{-12})$	–	$0.062\;(1.77\!\times\!10^{-3})$	–	$8.51\!\times\!10^{-8}\;(6.80\!\times\!10^{-8})$	–	$0.243\;(8.12\!\times\!10^{-3})$	–
$\mathcal{A}_{4}$	Best pairwise router	$1.23\!\times\!10^{-8}\;(1.75\!\times\!10^{-9})$	$<10^{-3}$	$0.057\;(2.07\!\times\!10^{-3})$	$<10^{-3}$	$7.24\!\times\!10^{-7}\;(8.62\!\times\!10^{-8})$	$<10^{-3}$	$0.069\;(2.50\!\times\!10^{-3})$	$<10^{-3}$	$6.09\!\times\!10^{-7}\;(7.81\!\times\!10^{-8})$	$<10^{-3}$	$0.279\;(7.70\!\times\!10^{-3})$	$<10^{-3}$
$\mathcal{A}_{4}$	Router $(\mathcal{A}_{4})$	$1.23\!\times\!10^{-13}\;(4.75\!\times\!10^{-15})$	–	$0.055\;(1.67\!\times\!10^{-3})$	–	$2.35\!\times\!10^{-13}\;(2.12\!\times\!10^{-14})$	–	$0.062\;(1.78\!\times\!10^{-3})$	–	$7.00\!\times\!10^{-8}\;(1.01\!\times\!10^{-8})$	–	$0.243\;(8.03\!\times\!10^{-3})$	–
$\mathcal{A}_{5}$	Best pairwise router	$1.23\!\times\!10^{-8}\;(1.75\!\times\!10^{-9})$	$<10^{-3}$	$0.057\;(2.07\!\times\!10^{-3})$	$<10^{-3}$	$6.95\!\times\!10^{-8}\;(9.13\!\times\!10^{-9})$	$<10^{-3}$	$0.070\;(2.91\!\times\!10^{-3})$	$<10^{-3}$	$6.09\!\times\!10^{-7}\;(7.81\!\times\!10^{-8})$	$<10^{-3}$	$0.279\;(7.70\!\times\!10^{-3})$	$<10^{-3}$

Appendix

Table 12: Relative residual at a fixed 25 ms cost-calibrated selected-action budget and time-averaged residual AUC for growing solver ensembles and their validation-selected best constituent pairwise routers (lower is better). Values are mean (standard error) over 64 shared test instances. We use

R(t)=\|f_{h}-\mathcal{L}_{h}^{a}u_{h}(t)\|_{2}/\|f_{h}\|_{2}

and

\mathrm{AUC}_{25}=(25\,\mathrm{ms})^{-1}\int_{0}^{25\,\mathrm{ms}}R(t)\,dt

. The state is held constant between completed actions; complete update and residual-refresh costs, the initial residual, and learned-router inference are charged. Each

p

-value is from a paired one-sided

t

-test with the alternative that the ensemble router is lower than its best pairwise comparator.

Jacobi-related solvers
	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
Method	Error	AUC	Error	AUC	Error	AUC
Solver Only	$2.53\!\times\!10^{-1}\;(7.91\!\times\!10^{-2})$	$2.82\!\times\!10^{-1}\;(8.81\!\times\!10^{-2})$	$3.41\!\times\!10^{-5}\;(1.39\!\times\!10^{-5})$	$\mathbf{1.39\!\times\!10^{-3}\;(5.66\!\times\!10^{-4})}$	$2.20\!\times\!10^{-13}\;(9.06\!\times\!10^{-14})$	$\mathbf{4.67\!\times\!10^{-5}\;(1.92\!\times\!10^{-5})}$
HINTS	$\mathbf{1.75\!\times\!10^{-8}\;(5.54\!\times\!10^{-9})}$	$1.57\!\times\!10^{-1}\;(4.93\!\times\!10^{-2})$	$4.62\!\times\!10^{-9}\;(1.30\!\times\!10^{-9})$	$1.49\!\times\!10^{-3}\;(6.06\!\times\!10^{-4})$	$3.42\!\times\!10^{-11}\;(8.64\!\times\!10^{-12})$	$4.71\!\times\!10^{-5}\;(1.94\!\times\!10^{-5})$
Learned Greedy	$1.91\!\times\!10^{-8}\;(6.15\!\times\!10^{-9})$	$\mathbf{7.83\!\times\!10^{-2}\;(2.45\!\times\!10^{-2})}$	$\mathbf{3.03\!\times\!10^{-10}\;(9.01\!\times\!10^{-11})}$	$1.81\!\times\!10^{-3}\;(7.39\!\times\!10^{-4})$	$\mathbf{1.16\!\times\!10^{-15}\;(3.42\!\times\!10^{-16})}$	$2.22\!\times\!10^{-4}\;(9.14\!\times\!10^{-5})$
GS-related solvers

Appendix

Table 13: Fourier-mode error at a 25 ms wall-clock budget for Poisson on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Bold marks the lowest deployable method within each solver family and column.

Jacobi-related solvers
	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
Method	Error	AUC	Error	AUC	Error	AUC
Solver Only	$8.21\!\times\!10^{-2}\;(2.57\!\times\!10^{-2})$	$8.78\!\times\!10^{-2}\;(2.75\!\times\!10^{-2})$	$2.19\!\times\!10^{-4}\;(8.95\!\times\!10^{-5})$	$1.82\!\times\!10^{-3}\;(7.43\!\times\!10^{-4})$	$9.26\!\times\!10^{-10}\;(3.81\!\times\!10^{-10})$	$\mathbf{7.17\!\times\!10^{-5}\;(2.95\!\times\!10^{-5})}$
HINTS	$8.12\!\times\!10^{-9}\;(2.69\!\times\!10^{-9})$	$6.22\!\times\!10^{-2}\;(1.95\!\times\!10^{-2})$	$3.49\!\times\!10^{-9}\;(1.22\!\times\!10^{-9})$	$1.81\!\times\!10^{-3}\;(7.37\!\times\!10^{-4})$	$1.21\!\times\!10^{-10}\;(4.03\!\times\!10^{-11})$	$7.20\!\times\!10^{-5}\;(2.96\!\times\!10^{-5})$
Learned Greedy	$\mathbf{5.98\!\times\!10^{-9}\;(1.70\!\times\!10^{-9})}$	$\mathbf{2.37\!\times\!10^{-2}\;(7.41\!\times\!10^{-3})}$	$\mathbf{6.84\!\times\!10^{-10}\;(1.99\!\times\!10^{-10})}$	$\mathbf{1.55\!\times\!10^{-3}\;(6.30\!\times\!10^{-4})}$	$\mathbf{3.14\!\times\!10^{-13}\;(1.04\!\times\!10^{-13})}$	$2.14\!\times\!10^{-4}\;(8.81\!\times\!10^{-5})$
GS-related solvers

Appendix

Table 14: Fourier-mode error at a 25 ms wall-clock budget for ConvDiff on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Bold marks the lowest deployable method within each solver family and column.

Jacobi-related solvers
	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
Method	Error	AUC	Error	AUC	Error	AUC
Solver Only	$5.19\!\times\!10^{-1}\;(1.63\!\times\!10^{-1})$	$5.68\!\times\!10^{-1}\;(1.78\!\times\!10^{-1})$	$1.74\!\times\!10^{-4}\;(7.08\!\times\!10^{-5})$	$3.32\!\times\!10^{-3}\;(1.35\!\times\!10^{-3})$	$2.15\!\times\!10^{-11}\;(8.82\!\times\!10^{-12})$	$\mathbf{1.13\!\times\!10^{-4}\;(4.65\!\times\!10^{-5})}$
HINTS	$1.30\!\times\!10^{-6}\;(4.32\!\times\!10^{-7})$	$3.34\!\times\!10^{-1}\;(1.05\!\times\!10^{-1})$	$1.52\!\times\!10^{-7}\;(3.90\!\times\!10^{-8})$	$3.34\!\times\!10^{-3}\;(1.36\!\times\!10^{-3})$	$1.59\!\times\!10^{-9}\;(4.52\!\times\!10^{-10})$	$1.14\!\times\!10^{-4}\;(4.67\!\times\!10^{-5})$
Learned Greedy	$\mathbf{6.83\!\times\!10^{-7}\;(1.80\!\times\!10^{-7})}$	$\mathbf{1.41\!\times\!10^{-1}\;(4.40\!\times\!10^{-2})}$	$\mathbf{3.83\!\times\!10^{-8}\;(1.01\!\times\!10^{-8})}$	$\mathbf{3.25\!\times\!10^{-3}\;(1.33\!\times\!10^{-3})}$	$\mathbf{4.10\!\times\!10^{-12}\;(1.35\!\times\!10^{-12})}$	$3.99\!\times\!10^{-4}\;(1.64\!\times\!10^{-4})$
GS-related solvers

Appendix

Table 15: Fourier-mode error at a 25 ms wall-clock budget for Anisotropic diffusion on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Bold marks the lowest deployable method within each solver family and column.

Actions	Method	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
		Error	AUC	Error	AUC	Error	AUC
$\mathcal{A}_{3}$	Best pairwise router	$5.69\!\times\!10^{-15}\;(2.58\!\times\!10^{-15})$	$0.012\;(5.41\!\times\!10^{-3})$	$1.05\!\times\!10^{-15}\;(4.13\!\times\!10^{-16})$	$1.52\!\times\!10^{-4}\;(7.32\!\times\!10^{-5})$	$1.75\!\times\!10^{-16}\;(7.88\!\times\!10^{-17})$	$5.19\!\times\!10^{-5}\;(2.77\!\times\!10^{-5})$
$\mathcal{A}_{3}$	Router $(\mathcal{A}_{3})$	$\mathbf{2.36\!\times\!10^{-15}\;(1.04\!\times\!10^{-15})}$	$\mathbf{0.012\;(5.40\!\times\!10^{-3})}$	$\mathbf{7.05\!\times\!10^{-18}\;(2.88\!\times\!10^{-18})}$	$\mathbf{1.51\!\times\!10^{-4}\;(7.29\!\times\!10^{-5})}$	$\mathbf{2.94\!\times\!10^{-18}\;(1.22\!\times\!10^{-18})}$	$\mathbf{5.19\!\times\!10^{-5}\;(2.78\!\times\!10^{-5})}$
$\mathcal{A}_{4}$	Best pairwise router	$\mathbf{5.69\!\times\!10^{-15}\;(2.58\!\times\!10^{-15})}$	$\mathbf{0.012\;(5.41\!\times\!10^{-3})}$	$1.05\!\times\!10^{-15}\;(4.13\!\times\!10^{-16})$	$1.52\!\times\!10^{-4}\;(7.32\!\times\!10^{-5})$	$1.75\!\times\!10^{-16}\;(7.88\!\times\!10^{-17})$	$\mathbf{5.19\!\times\!10^{-5}\;(2.77\!\times\!10^{-5})}$
$\mathcal{A}_{4}$	Router $(\mathcal{A}_{4})$	$5.98\!\times\!10^{-15}\;(2.94\!\times\!10^{-15})$	$0.012\;(5.41\!\times\!10^{-3})$	$\mathbf{4.01\!\times\!10^{-17}\;(1.85\!\times\!10^{-17})}$	$\mathbf{1.52\!\times\!10^{-4}\;(7.31\!\times\!10^{-5})}$	$\mathbf{3.16\!\times\!10^{-18}\;(1.51\!\times\!10^{-18})}$	$5.21\!\times\!10^{-5}\;(2.78\!\times\!10^{-5})$
$\mathcal{A}_{5}$	Best pairwise router	$5.69\!\times\!10^{-15}\;(2.58\!\times\!10^{-15})$	$\mathbf{0.012\;(5.41\!\times\!10^{-3})}$	$1.05\!\times\!10^{-15}\;(4.13\!\times\!10^{-16})$	$\mathbf{1.52\!\times\!10^{-4}\;(7.32\!\times\!10^{-5})}$	$1.75\!\times\!10^{-16}\;(7.88\!\times\!10^{-17})$	$\mathbf{5.19\!\times\!10^{-5}\;(2.77\!\times\!10^{-5})}$

Appendix

Table 16: Fourier-mode error at a 25 ms wall-clock budget for Poisson growing solver ensembles on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Each ensemble router is compared with its validation-selected best constituent pairwise router; bold marks the lower method within each ensemble size and column.

Actions	Method	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
		Error	AUC	Error	AUC	Error	AUC
$\mathcal{A}_{3}$	Best pairwise router	$4.47\!\times\!10^{-9}\;(1.73\!\times\!10^{-9})$	$\mathbf{1.97\!\times\!10^{-3}\;(5.03\!\times\!10^{-4})}$	$1.14\!\times\!10^{-9}\;(6.59\!\times\!10^{-10})$	$\mathbf{1.45\!\times\!10^{-4}\;(7.93\!\times\!10^{-5})}$	$3.76\!\times\!10^{-12}\;(1.66\!\times\!10^{-12})$	$\mathbf{2.71\!\times\!10^{-5}\;(1.45\!\times\!10^{-5})}$
$\mathcal{A}_{3}$	Router $(\mathcal{A}_{3})$	$\mathbf{6.04\!\times\!10^{-16}\;(2.38\!\times\!10^{-16})}$	$1.98\!\times\!10^{-3}\;(5.05\!\times\!10^{-4})$	$\mathbf{2.64\!\times\!10^{-17}\;(1.28\!\times\!10^{-17})}$	$1.47\!\times\!10^{-4}\;(8.03\!\times\!10^{-5})$	$\mathbf{1.12\!\times\!10^{-18}\;(5.09\!\times\!10^{-19})}$	$2.73\!\times\!10^{-5}\;(1.47\!\times\!10^{-5})$
$\mathcal{A}_{4}$	Best pairwise router	$1.64\!\times\!10^{-10}\;(8.82\!\times\!10^{-11})$	$\mathbf{1.97\!\times\!10^{-3}\;(5.00\!\times\!10^{-4})}$	$9.04\!\times\!10^{-11}\;(4.16\!\times\!10^{-11})$	$1.47\!\times\!10^{-4}\;(7.97\!\times\!10^{-5})$	$2.49\!\times\!10^{-11}\;(1.48\!\times\!10^{-11})$	$2.75\!\times\!10^{-5}\;(1.46\!\times\!10^{-5})$
$\mathcal{A}_{4}$	Router $(\mathcal{A}_{4})$	$\mathbf{6.02\!\times\!10^{-16}\;(2.25\!\times\!10^{-16})}$	$1.97\!\times\!10^{-3}\;(5.03\!\times\!10^{-4})$	$\mathbf{3.78\!\times\!10^{-17}\;(1.82\!\times\!10^{-17})}$	$\mathbf{1.45\!\times\!10^{-4}\;(7.89\!\times\!10^{-5})}$	$\mathbf{7.55\!\times\!10^{-19}\;(2.03\!\times\!10^{-19})}$	$\mathbf{2.71\!\times\!10^{-5}\;(1.44\!\times\!10^{-5})}$
$\mathcal{A}_{5}$	Best pairwise router	$7.28\!\times\!10^{-10}\;(3.57\!\times\!10^{-10})$	$1.97\!\times\!10^{-3}\;(5.04\!\times\!10^{-4})$	$6.28\!\times\!10^{-11}\;(3.35\!\times\!10^{-11})$	$1.47\!\times\!10^{-4}\;(7.99\!\times\!10^{-5})$	$5.99\!\times\!10^{-11}\;(3.55\!\times\!10^{-11})$	$2.79\!\times\!10^{-5}\;(1.48\!\times\!10^{-5})$

Appendix

Table 17: Fourier-mode error at a 25 ms wall-clock budget for convection–diffusion growing solver ensembles on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Each ensemble router is compared with its validation-selected best constituent pairwise router; bold marks the lower method within each ensemble size and column.

Actions	Method	Mode $(1,1)$		Mode $(5,5)$		Mode $(10,10)$
		Error	AUC	Error	AUC	Error	AUC
$\mathcal{A}_{3}$	Best pairwise router	$\mathbf{2.56\!\times\!10^{-15}\;(6.66\!\times\!10^{-16})}$	$\mathbf{0.047\;(0.012)}$	$1.26\!\times\!10^{-12}\;(6.90\!\times\!10^{-13})$	$2.42\!\times\!10^{-3}\;(1.32\!\times\!10^{-3})$	$\mathbf{2.25\!\times\!10^{-17}\;(6.13\!\times\!10^{-18})}$	$3.99\!\times\!10^{-4}\;(2.13\!\times\!10^{-4})$
$\mathcal{A}_{3}$	Router $(\mathcal{A}_{3})$	$1.90\!\times\!10^{-14}\;(9.20\!\times\!10^{-15})$	$0.047\;(0.012)$	$\mathbf{1.15\!\times\!10^{-16}\;(5.16\!\times\!10^{-17})}$	$\mathbf{2.30\!\times\!10^{-3}\;(1.32\!\times\!10^{-3})}$	$2.35\!\times\!10^{-17}\;(6.25\!\times\!10^{-18})$	$\mathbf{3.84\!\times\!10^{-4}\;(2.14\!\times\!10^{-4})}$
$\mathcal{A}_{4}$	Best pairwise router	$2.69\!\times\!10^{-15}\;(7.08\!\times\!10^{-16})$	$\mathbf{0.047\;(0.012)}$	$1.56\!\times\!10^{-11}\;(6.59\!\times\!10^{-12})$	$2.42\!\times\!10^{-3}\;(1.32\!\times\!10^{-3})$	$1.51\!\times\!10^{-11}\;(7.02\!\times\!10^{-12})$	$4.00\!\times\!10^{-4}\;(2.14\!\times\!10^{-4})$
$\mathcal{A}_{4}$	Router $(\mathcal{A}_{4})$	$\mathbf{2.48\!\times\!10^{-15}\;(6.38\!\times\!10^{-16})}$	$0.047\;(0.012)$	$\mathbf{1.88\!\times\!10^{-12}\;(1.09\!\times\!10^{-12})}$	$\mathbf{2.30\!\times\!10^{-3}\;(1.33\!\times\!10^{-3})}$	$\mathbf{2.51\!\times\!10^{-15}\;(9.32\!\times\!10^{-16})}$	$\mathbf{3.84\!\times\!10^{-4}\;(2.14\!\times\!10^{-4})}$
$\mathcal{A}_{5}$	Best pairwise router	$2.69\!\times\!10^{-15}\;(7.08\!\times\!10^{-16})$	$0.047\;(0.012)$	$1.56\!\times\!10^{-11}\;(6.59\!\times\!10^{-12})$	$2.42\!\times\!10^{-3}\;(1.32\!\times\!10^{-3})$	$1.51\!\times\!10^{-11}\;(7.02\!\times\!10^{-12})$	$4.00\!\times\!10^{-4}\;(2.14\!\times\!10^{-4})$

Appendix

Table 18: Fourier-mode error at a 25 ms wall-clock budget for anisotropic diffusion growing solver ensembles on 64 shared test instances. Each entry is mean (standard error) of

|\widehat{e}_{k,k}|

, or its time-averaged wall-clock AUC, using NumPy’s unnormalized DFT of the demeaned solution error. Lower is better. Each ensemble router is compared with its validation-selected best constituent pairwise router; bold marks the lower method within each ensemble size and column.

Figure 10: Learned-router solver usage for paired classical–neural solvers across 64 test instances at a relative solution-error tolerance of

10^{-12}

. Each panel shows the selected action at each decision epoch, with instances grouped by their complete action sequence. Gray indicates that the tolerance has been reached, while red denotes censored instances that do not reach the tolerance within the evaluation horizon.

Figure 11: Solver usage by the learned router for growing Poisson solver ensembles on 128 test instances. Rows correspond to instances sorted by solve length, and colors indicate the executed primitive actions over normalized solve progress. The right column reports the fraction of instances using each action at each point in the solve.

Figure 12: Solver usage by the learned router for growing convection-diffusion solver ensembles on 128 test instances. Rows correspond to instances sorted by solve length, and colors indicate the executed primitive actions over normalized solve progress. The right column reports the fraction of instances using each action at each point in the solve.

Figure 13: Solver usage by the learned router for growing anisotropic diffusion solver ensembles on 128 test instances. Rows correspond to instances sorted by solve length, and colors indicate the executed primitive actions over normalized solve progress. The right column reports the fraction of instances using each action at each point in the solve.

	Poisson		ConvDiff
Method	Time (ms)	$p$ -value	Time (ms)	$p$ -value
Jacobi-related solvers
Jacobi Only	–	–	–	–
HINTS-Jacobi ( $m=50/50$ )	2999.34 (1822.99) [14/16]	–	948.34 (302.28) [12/16]	–
Learned Greedy-Jacobi	302.78 (71.31) [12/16]	–	878.31 (305.26) [12/16]	–
True-Greedy-Jacobi	2780.05 (1710.63) [14/16]	–	859.51 (293.84) [12/16]	–

Appendix

Table 19: Selected-action wall-clock replay time to relative solution error

10^{-6}

on the

512\times 512

grid (lower is better). Values are mean (standard error) in milliseconds over successful cases among 16 shared test instances; entries without brackets have 16/16 successes. Brackets report incomplete coverage, and a dash indicates no successes. The reported HINTS result uses the best observed cadence in

\{5,10,25,50\}

for each equation and solver family at this tolerance. One-sided paired

t

-tests compare each full-coverage deployable baseline with the learned router under the alternative that the router is faster; tests are omitted when either method lacks full coverage. Bold marks the fastest deployable method with full coverage, while True-Greedy is an oracle with access to the exact error.

	Poisson		ConvDiff		Aniso
Method	Time (ms)	$p$ -value	Time (ms)	$p$ -value	Time (ms)	$p$ -value
Jacobi-related solvers
Jacobi Only	–	–	1116.25 (8.57)	$<10^{-3}$	–	–
HINTS-Jacobi ( $m=10/10/50$ )	599.86 (177.87)	–	583.50 (109.39)	$<10^{-3}$	454.80 (78.08)	$<10^{-3}$
Learned Greedy-Jacobi	124.03 (37.49) [62/64]	–	148.24 (23.96)	–	314.83 (52.80)	–
True-Greedy-Jacobi	148.39 (40.34)	–	144.40 (23.41)	–	299.83 (49.03)	–

Appendix

Table 20: Selected-action wall-clock replay time to relative solution error

10^{-6}

with zero Dirichlet boundary conditions (lower is better). Values are mean (standard error) in milliseconds over successful cases among 64 shared test instances with

128\times 128

interior unknowns; entries without brackets have 64/64 successes. Brackets report incomplete coverage, and a dash indicates no successes. The reported HINTS result uses the best tested cadence in

\{5,10,25,50\}

for each equation and solver family on these test instances and is therefore an optimistic comparator. One-sided paired

t

-tests compare each full-coverage deployable baseline with the learned router under the alternative that the router is faster; tests are omitted when either method lacks full coverage. Bold marks the fastest deployable method with full coverage, while True-Greedy is an oracle with access to the exact error.

A Greedy PDE Router for Blending Neural Operators and Classical Methods

Abstract

Figures & tables

Appendix

PDE Solving

Transferability of Learned States in Neural PDE Solvers

Error-Conditioned Neural Solvers

Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

Jacobi-related Solvers
Equation	Poisson	Conv	Aniso
Methods	Time (ms)	Time (ms)	Time (ms)
Jacobi Only	6428.52 (4.15)	10146.76 (14.21)	–
HINTS-Jacobi ( $m=50/50/50$ )	2288.27 (125.46)	3278.90 (161.52)	1845.77 (126.04)
Learned Greedy-Jacobi	1280.80 (64.97)	2900.52 (181.73)	1117.90 (77.21) [62/64]
True-Greedy-Jacobi	1226.02 (66.29)	2120.53 (103.75)	1130.38 (75.91)

Abstract

PDE Solving

Explore similar work

Transferability of Learned States in Neural PDE Solvers

Error-Conditioned Neural Solvers

Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations