A Greedy PDE Router for Blending Neural Operators and Classical Methods

Sahana Rayan; Yash Patel; Ambuj Tewari

stat.MESep 29, 2025

A Greedy PDE Router for Blending Neural Operators and Classical Methods

Authors: Sahana Rayan, Yash Patel, Ambuj Tewari

Organizations: Department of Statistics University of Michigan Ann Arbor, MI 48104, USA

Abstract

When solving PDEs, classical numerical solvers are often computationally expensive, while machine learning methods can suffer from spectral bias, failing to capture high-frequency components. Designing an optimal hybrid iterative solver--where, at each iteration, a solver is selected from an ensemble of solvers to leverage their complementary strengths--poses a challenging combinatorial problem. While greedy selection is desirable for its constant-factor approximation guarantee to the optimal solution under Lipschitz assumptions, it requires knowledge of the true error at each step, which is unavailable in practice. We address this by proposing an approximate greedy router that efficiently mimics a greedy approach to solver selection. Empirical results on the Poisson, convection-diffusion, and anisotropic diffusion equations show that our method consistently reduces time-to-solution relative to single-solver baselines and existing hybrid approaches such as HINTS.

Figures & tables

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)

Table 1: Selected-action wall-clock replay time in milliseconds (lower is better), reported as mean (standard error) in milliseconds over successful cases. Brackets indicate the number of successes when coverage is incomplete, and a dash indicates no successful cases. Bold marks the fastest deployable method with full coverage in each solver family; True-Greedy is an oracle.

Figure 1: Convergence histories for representative test instances using Jacobi. Columns: Poisson, Convection Diffusion, and anisotropic Diffusion. Greedy yields the fastest monotone decay.

Figure 2: Action usage of the learned Jacobi router across 64 test instances. Bars show the fraction assigned to the neural action, the classical action, or already converged at each of the first 20 routing rounds. Results for other solvers appear in Section F.6 .

Actions	Method	Poisson		ConvDiff		Aniso
Actions	Method	Time (ms)	$p$ -value	Time (ms)	$p$ -value	Time (ms)	$p$ -value
$\mathcal{A}_{3}$	Best pairwise router	9.21 (0.69)	–	18.56 (0.83)	–	16.92 (0.22)	–
$\mathcal{A}_{3}$	Router $(\mathcal{A}_{3})$	4.33 (0.27)	$<10^{-3}$	6.99 (0.30)	$<10^{-3}$	14.45 (0.37)	$<10^{-3}$
$\mathcal{A}_{4}$	Best pairwise router	9.21 (0.69)	–	16.76 (0.76)	–	14.93 (0.32)	–
$\mathcal{A}_{4}$	Router $(\mathcal{A}_{4})$	4.33 (0.27)	$<10^{-3}$	7.05 (0.30)	$<10^{-3}$	14.60 (0.38)	$0.057$
$\mathcal{A}_{5}$	Best pairwise router	9.21 (0.69)	–	14.09 (0.65)	–	14.93 (0.32)	–

Table 2: Selected-action wall-clock replay time for growing solver ensembles, reported in milliseconds as mean (standard error) over successful cases. The pairwise comparator is selected on validation data. Brackets indicate incomplete coverage, a dash indicates no successes, and one-sided paired-test

p

-values are reported for full-coverage comparisons.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

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}|

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}|

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

PDE Solving

PDE solving develops numerical and learned methods to approximate solutions of partial differential equations across scientific applications.

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Oct 7, 2026

Transferability of Learned States in Neural PDE Solvers

Assessing useful reuse in neural PDE solvers is challenging: final accuracy can reflect source learning and target-time computation. Our reuse contract separates solution accuracy, learning contribution, and numerical utility through paired state comparisons, matched target information and budgets, and cost accounting. A literature audit extracts 18 version-specific protocol records from 12 papers, documenting retained states, target-time resources, and reported controls. For a fixed linear system and residual tolerance, we construct two initial guesses with identical solution-error, energy-error, and residual norms, reaching the same solution with different conjugate-gradient (CG) iteration counts. Across 240 source-training trajectories, two linear PDE families, Fourier neural operators and convolutional networks, a fixed predictor's benefit reverses across correction algorithms. Among pairs with both relative prediction errors less than or equal to 5 percent on 64 in-distribution tasks (63 by 63 interior grids), reductions in all three norms accompany more CG iterations, at mean taskwise rates of 23.5 percent and 23.9 percent in two libraries. Work-based selection saves 2.50-3.33 CG iterations on held-out in-distribution tasks; matched adaptation demonstrates finite-budget pretraining value. Independent batches confirm a 0.73 percent complete online saving for one physics-trained Fourier neural operator against zero-initialized Poisson-preconditioned CG. Reuse requires matched state comparisons and downstream computational evidence.

Shunye Wang, Haochen Wen, Shuo Li Liu +3

University of Chicago · University of Cambridge · Princeton University +2

Neural PDE Solvers PDE Surrogate Modeling

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Jun 25, 2026

Error-Conditioned Neural Solvers

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

10\times

on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Haina Jiang, Liam Wang, Peng-Chen Chen +4

University of Michigan · KAIST AI · Los Alamos National Laboratory

Neural PDE Solvers PDE Surrogate Modeling

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May 6, 2026

Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains

\mathcal{O}(τ)

, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies

C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)

, where

\varepsilon_{net}

measures the network approximation quality and

δ

the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics Tianjin University Tianjin, 300072, China · School of Mechanical and Aerospace Engineering, Jilin University Jilin University Changchun, 130025, China

Neural PDE Solvers PDE Solving Neural Operators Numerical Integration

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Abstract

PDE Solving

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