Lossy Compression of PDE Training Inputs: Field Reconstruction Error Does Not Order the Cost to a Trained Operator

Huy Hoang Le

cs.LGOct 5, 2026

Lossy Compression of PDE Training Inputs: Field Reconstruction Error Does Not Order the Cost to a Trained Operator

Authors: Huy Hoang Le

Organizations: Powermore Ltd., Da Nang, Vietnam

Abstract

Operator-learning benchmarks are stored at full precision and have grown to terabyte scale. Rate-distortion theory says how many bits the stored field needs, while a practitioner needs to know how accurate an operator trained on the compressed data will be. We show that the first does not determine the second, and measure why, compressing the input fields while targets and test inputs stay at full precision. A solution operator attenuates a perturbation of its input. Pushing a compressed field through a surrogate already trained at full precision measures how much of the perturbation that surrogate transmits. The fraction is consistent with the smoothing behaviour of the underlying equation, and it spans more than two orders of magnitude across PDE families. Field reconstruction error is computed before the attenuation and cannot see it. For operators trained with mean squared error it inverts 36 of 104 cost comparisons across datasets, where a probe built from the same forward passes inverts 12. Two families that PDEBench stores with identical initial conditions differ threefold downstream at identical field error. Under the relative-L2 objective of the reference recipe the separation narrows, while the ordering of the family-level median transmission factors is unchanged. After one full-precision training run, the probe evaluates an entire rate curve by forward passes alone. It ranks datasets and rates consistently across the codecs and architectures we test, while its magnitude does not transfer between them.

Figures & tables

Figure 1: The two quantities the paper compares, and the probe that connects them. (a) Field error is measured before training, downstream cost after it. (b) The probe uses forward passes through one model trained at full precision, so an entire rate curve costs that single training run and no training on compressed data.

family	field error	fp32 error	cost	seeds
advection	$2.743\%$	$0.855\pm 0.010\%$	$\mathbf{3.03\pm 0.21\times}$	$5$
Burgers	$2.743\%$	$9.275\pm 0.022\%$	$\mathbf{1.01\pm 0.00\times}$	$2$
reaction–diffusion	$1.807\%$	$0.097\pm 0.000\%$	$\mathbf{1.05\pm 0.00\times}$	$2$

Table 1: Same codec, same rate (

0.25

bits per value,

128\times

compression), three PDE families, on the corrected codebook. The first two rows share their inputs bit for bit, and so their field error, and differ by

3.0\times

downstream.

family	$T$ / field error	the solution operator approximated
advection	$0.97$ to $1.12$	a shift, transmits everything
Burgers	$0.64$ to $0.93$	nonlinear, transmits most
shallow water 2D	$0.35$ to $0.42$	hyperbolic with a source
Darcy 2D	$0.12$ to $0.15$	elliptic solve, smooths hard
reaction–diffusion	$0.032$ to $0.084$	close to a projection
reaction–diffusion, far $\nu,\rho$	$\mathbf{0.007}$ to $0.030$	the strongest projection here

Table 2: How much of an input perturbation the trained surrogate of each solution operator passes to its output, over all sixteen trained points. The spread is

\mathbf{160\times}

and the ordering is consistent with the spectral picture described above. This is measured by forward passes through one full-precision model, so it needs no training on compressed data, and it does not involve

\varepsilon_{0}

, so none of the confounds discussed in Section 5 and Appendix G apply to it.

Figure 2: The sixteen trained measurements of Table 5 , the same measured cost on both vertical axes and the two candidate predictors on the horizontal, points of each family joined. Left, field error: every family is individually monotone and the families are offset from one another, and that offset puts 36 of the 104 across-family pairs the wrong way. Right, the probe brings them onto one line, twelve across and none within. Points below the diagonal are over-estimates by a probe that still ranks correctly, the distinction Section 13 draws. All sixteen use the corrected codebook and mean-squared-error training.

family	dim	fp32 error	$r^{\star}(1.10)$	compression
Burgers	1	$9.275\%$	$0.1217$ bpv	$\mathbf{263\times}$
Darcy	2	$4.215\%$	$0.1647$ bpv	$\mathbf{194\times}$
reaction–diffusion	1	$0.097\%$	$0.4080$ bpv	$\mathbf{78\times}$
advection	1	$0.857\%$	$>8$ bpv	none at $10\%$

Table 3: The candidate rate nominated by the probe as the smallest at which the trained operator may remain within

10\%

of its full-precision accuracy. These are candidates, and the back-check below finds them in error in both directions, by up to a factor of two in stored bits. Advection never reaches this tolerance: its probe curve flattens at

1.11\times

, where the cap of 14 bits per coefficient leaves

0.36\%

field error however many bits are spent (Appendix L ).

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

family	$D$	samples	side info	stored rate at the lowest trained rate
1D families	$1024$	$10^{4}$	$6.6$ bits	$0.0689$ against $0.0625$ nominal, $1.10\times$
Darcy 2D	$4096$	$10^{4}$	$26.2$ bits	$0.0689$ against $0.0625$ , $1.10\times$
shallow water 2D	$4096$	$10^{3}$	$262.1$ bits	$\mathbf{0.0953}$ against $0.03125$ , $\mathbf{3.05\times}$

Appendix

Table 4: What the decoder-side mean and standard deviation vectors cost, charged as Section 3.1 requires. The first two rows share a ratio because Equation ( 5 ) does not depend on

D

; shallow water pays most because it has ten times fewer samples to amortise over. This column is the

2b_{\mathrm{f}}/n

part of Equation ( 5 ); adding the allocation table gives the

0.0068

bits per value that Section 7 charges. Rates in Table 5 and Figure 5 are nominal payload rates; Sections 7 and 8 charge the total.

family	rate (bpv)	field error	measured cost
advection	$4.0$	$0.379\%$	$1.19\times$
reaction–diffusion	$0.5$	$0.673\%$	$1.00\times$
advection	$1.0$	$0.762\%$	$1.52\times$
reaction–diffusion	$0.125$	$4.278\%$	$2.57\times$
Burgers	$0.125$	$5.746\%$	$1.07\times$
Darcy 2D	$0.25$	$13.540\%$	$\mathbf{1.27\times}$

Appendix

Table 5: Every trained measurement, six families, all on the corrected codebook and two seeds per cell. The first block holds the four families of the original study and the others the two added later; the last is reaction–diffusion at

\nu=0.5

\rho=10

, a second dataset of an equation already present rather than a new equation. Reaction–diffusion at

4.278\%

field error costs

2.57\times

; Burgers at

35.082\%

, a field reconstruction

8.2\times

worse, costs

2.28\times

family pair	point pairs	field error inverts	the probe inverts
advection $\times$ shallow water	$8$	$\mathbf{0}$	$0$
Burgers $\times$ Darcy 2D	$4$	$\mathbf{0}$	$1$
reaction–diffusion $\times$ far $\nu,\rho$	$8$	$\mathbf{0}$	$1$
advection $\times$ far $\nu,\rho$	$8$	$1$	$1$
reaction–diffusion $\times$ shallow water	$8$	$1$	$1$
advection $\times$ reaction–diffusion	$4$	$1$	$0$

Appendix

Table 6: The 15 family pairs, sorted by how often field error inverts them. It is exact on three pairs and wrong on 6 or 7 of 8 for two others: the information is present and its sign is not stable. Counting one vote per block rather than one per point pair, field error loses 4 blocks and the probe loses none.

Figure 3: At each of the seven rates all six families share, how many of the 15 family pairs field error orders the wrong way, against the

7.5

that chance would give. It inverts

9

at a sixteenth of a bit per value and at least

4

at every rate, so the failure is not an artefact of the rates we happened to train at. Costs here are the probe curve and field errors those of the corrected codebook, since training every family at every rate is the sweep this paper argues is unaffordable; the claim rests on the trained points of Table 5 and Figure 2 , not on this figure.

predictor	within a family	across families	$\rho$	worst magnitude
field reconstruction error	$0$ of $16$	$36$ of $104$	$+0.48$	—
$\sqrt{1+(\text{field error}/\varepsilon_{0})^{2}}$	$0$ of $16$	$29$ of $104$	$+0.66$	$71.2\times$
$\varepsilon_{0}$ alone, no codec information	$0$ of $16$	$38$ of $104$	$+0.31$	—
transmitted error $T$ alone	$0$ of $16$	$53$ of $104$	$+0.13$	—
the probe, $\sqrt{1+(T/\varepsilon_{0})^{2}}$	$0$ of $16$	$\mathbf{12}$ of $104$	$\mathbf{+0.93}$	$\mathbf{2.11\times}$

Appendix

Table 7: Taking the probe apart on the 16 trained measurements. Every predictor scores

0

16

within a family, because field error and cost are both monotone in the rate, so that column is a property of the sweep and not a result. The across-family column is the one that discriminates.

T

alone is at chance and

\varepsilon_{0}

alone does about as well as field error; only their ratio orders the datasets.

families	pairs	field error inverts	the probe inverts	blocks lost
all six	$104$	$36$ ( $35\%$ )	$\mathbf{12}$ ( $12\%$ )	$4$ vs $\mathbf{0}$ of $15$
without advection	$76$	$31$ ( $41\%$ )	$9$ ( $12\%$ )	$4$ vs $0$ of $10$
without Burgers	$76$	$22$ ( $29\%$ )	$9$ ( $12\%$ )	$2$ vs $0$ of $10$
without reaction–diffusion	$76$	$30$ ( $39\%$ )	$10$ ( $13\%$ )	$4$ vs $0$ of $10$
without Darcy 2D	$76$	$20$ ( $26\%$ )	$8$ ( $11\%$ )	$2$ vs $0$ of $10$
without shallow water	$56$	$20$ ( $36\%$ )	$6$ ( $11\%$ )	$2$ vs $0$ of $10$

Appendix

Table 8: The across-family comparison of Figure 2 , under mean squared error, with families removed: each in turn, the two that share initial conditions, and the two added last, of which shallow water trains on a tenth of the data and the far-parameter set repeats an equation. Blocks lost counts the family pairs on which a predictor inverts more than half of the point pairs, field error first. Neither predictor inverts a within-family pair in any row.

objective	field error	the probe	blocks lost	$\kappa$ order	worst $\rho$
mean squared error	$36$ of $104$	$12$ of $104$	$4$ vs $0$	—	$2.11\times$
relative $L^{2}$ , normalised targets	$23$ of $104$	$19$ of $104$	$1$ vs $0$	$1.00$	$2.88\times$
relative $L^{2}$ , physical field	$20$ of $104$	$\mathbf{11}$ of $104$	$1$ vs $0$	$1.00$	$1.72\times$

Appendix

Table 9: The sixteen trained cells under three training objectives, everything else fixed. Blocks lost counts the family pairs on which a predictor inverts more than half of the point pairs, field error first.

\kappa

order is the rank correlation of each family’s median transmission factor with its value under mean squared error, and worst

\rho

is Equation ( 11 ) over the sixteen cells. Within a family neither predictor inverts a pair, except one each under the normalised form.

family	mean squared error	relative $L^{2}$ , normalised	relative $L^{2}$ , physical
advection	$0.8570\%$	$0.7267,0.7256\%$	$0.7323,0.7193\%$
Burgers	$9.2747\%$	$3.4380,3.4438\%$	$3.4880,3.6654\%$
reaction–diffusion	$0.0972\%$	$\mathbf{0.0847,0.0581\%}$	$0.0846,0.0856\%$
Darcy 2D	$4.2154\%$	$2.3740,2.3652\%$	$2.4750,2.4803\%$
shallow water 2D	$0.4099\%$	$0.2903,0.3017\%$	$0.4027,0.3852\%$
reaction–diffusion, far $\nu,\rho$	$0.0207\%$	$\mathbf{0.0452,0.0304\%}$	$0.0260,0.0265\%$

Appendix

Table 10: Full-precision test error under the three objectives: for mean squared error the mean of two seeds, for relative

L^{2}

each seed. Bold marks the two families whose seeds the normalised form left far apart.

family	rate	field error	cost, MSE	normalised	physical	probe
advection	$1.0$	$0.762\%$	$1.52\times$	$1.12\times$	$1.11\times$	$1.42\times$
advection	$4.0$	$0.379\%$	$1.19\times$	$1.03\times$	$1.03\times$	$1.16\times$
Burgers	$0.03125$	$35.082\%$	$2.28\times$	$6.41\times$	$5.93\times$	$9.96\times$
Burgers	$0.125$	$5.746\%$	$1.07\times$	$1.64\times$	$1.62\times$	$1.83\times$
reaction–diffusion	$0.125$	$4.278\%$	$2.57\times$	$1.75\times$	$2.49\times$	$4.30\times$
reaction–diffusion	$0.5$	$0.673\%$	$1.00\times$	$1.14\times$	$0.98\times$	$1.03\times$

Appendix

Table 11: Every cell under the three objectives. Field error does not depend on the objective and agrees to three decimals across the three runs. Each cost is relative to the full-precision model of its own objective.

Figure 4: Left: eigenvalue spectrum of the output covariance,

4000

samples per family, normalised by the leading eigenvalue. Right: how many modes hold

99\%

of the energy, before and after the solution operator. Advection is an isometry and its two markers coincide. Reaction–diffusion and Darcy collapse onto a small fraction of the modes their inputs occupy; Burgers expands.

family	dim	$\gamma_{\text{in}}$	$\gamma_{\text{out}}$	eff. rank in	out	contraction	$r^{\star}(1.10)$
advection	1	$5.165$	$5.442$	$2.9$	$2.9$	$0.9999$	$>\!8$
Burgers	1	$5.165$	$2.232$	$2.9$	$1.3$	$0.4577$	$0.1217$
reaction–diffusion	1	$5.124$	$5.037$	$10.8$	$1.0$	$0.0923$	$0.4080$
Darcy	2	$1.562$	$2.473$	$8.3$	$1.4$	$0.1745$	$0.1647$

Appendix

Table 12: Structural quantities and the candidate rate. Ordering by contraction gives

0.9999,0.4577,0.1745,0.0923

against

r^{\star}

>\!8,0.1217,0.1647,0.4080

, which is not monotone. With four families and the confound described below we report the table and fit nothing to it.

family	fp32 error (%)	$r^{\star}(1.05)$	$r^{\star}(1.10)$	$r^{\star}(1.25)$
advection	$0.8570$	$>\!8$	$>\!8$	$1.2800$
Burgers	$9.2747$	$0.1604$	$0.1217$	$0.1048$
reaction–diffusion	$0.0972$	$0.4659$	$0.4080$	$0.2834$
Darcy	$4.2154$	$0.3954$	$0.1647$	$0.0578$

Appendix

Table 13:

r^{\star}

in bits per value at three tolerances, on the corrected probe curves. Advection reaches only the loosest, because its curve flattens at

1.11\times

Figure 5: Predicted against measured cost at all 48 points, grouped by codec because the paper claims something different about each. Circles are our transform codec across six families, triangles the Daubechies-4 wavelet, open squares the cross-codec arms. The shaded band is

1.6\times

, the worst case the first four families gave. Two arms are left out: 2D diffusion–reaction and the velocity channel of 1D computational fluid dynamics (CFD), whose pre-registered feasibility gate found every measured cost at

1.00\times

, so they would pile up at the origin and make the probe look exact on data that cannot test it.

Figure 6: One advection sample at

0.25

bits per value under both codecs, with the error-bounded tolerance tuned to match the transform codec’s field error to two decimal places. Top: the reconstructions. Middle: the errors in space, on a shared vertical scale. Bottom: the same two errors per DCT coefficient, against the signal itself. The transform error tracks the signal and falls with it; the error-bounded error does not.

our transform codec, for reference
family	arm	rate (bpv)	field error	probe	measured	ratio
advection	transform	$0.2500$	$2.743\%$	$2.98\times$	$3.03\pm 0.21\times$	$0.99$
reaction–diffusion	transform	$0.1250$	$4.278\%$	$3.81\times$	$2.58\pm 0.10\times$	$1.48$
SZ3, an error-bounded compressor
advection	$35\%$ target	$0.1009$	$34.49\%$	$29.49\times$	$12.39\pm 0.86\times$	$2.38$
advection	$12\%$ target	$0.2468$	$11.87\%$	$7.17\times$	$1.54\pm 0.02\times$	$\mathbf{4.65}$

Appendix

Table 14: The probe measured inside two codecs that are not ours, against the two control rows it was calibrated on. The error-bounded arm here is the Lorenzo stand-in we wrote, not the shipping SZ3 that Section 7 runs; what it establishes is that the probe over-predicts inside a per-sample bounded-error codec, and Section 7 shows the stand-in was if anything generous to SZ3 on rate. Transform arms use five seeds, SZ3 arms three and wavelet arms two; where a standard deviation is given it is over seeds. Every 1 of the 12 non-control rows over-predicts. The pre-registered bars were

2.0\times

for SZ3, missed at

4.65\times

, and

1.61\times

worst with a

1.25\times

median for the wavelet, missed at

3.49\times

and

2.14\times

Figure 7: Downstream cost against stored bits per value, four families, each with its own marker so that identity does not rest on colour alone. Lines are the probe curve, hollow marks are trained measurements, triangles mark

r^{\star}(1.10)

family	probe $r^{\star}$	trained fit	ratio	the check
Burgers	$0.1217$	$0.1037$	$0.85$	$0.125$ bpv costs $1.07\times$ , consistent
reaction–diffusion	$0.4080$	$0.2234$	$0.55$	$0.25$ bpv, below it, costs $1.05\times$ : conservative
Darcy 2D	$0.1647$	—	—	$0.25$ bpv costs $\mathbf{1.27\times}$ , outside
shallow water 2D	$3.349$	$6.836$	$2.04$	extrapolated, directional only
reaction–diffusion, far	$0.4130$	$0.7006$	$1.70$	$0.5$ bpv costs $\mathbf{1.15\times}$ , outside
advection	$>\!8$	—	—	$1$ and $4$ bpv cost $1.52$ and $1.19\times$ , consistent

Appendix

Table 15: The back-check. The trained fit is where a line through

\log(\mathrm{cost}-1)

against

\log r

over a family’s trained cells crosses the tolerance, computed where a family has at least three cells and one of them outside the tolerance; the ratio is the fit over the probe’s rate. The nominated rate is optimistic on Darcy, shallow water and the far-parameter set, and conservative on reaction–diffusion, where a rate

1.6\times

below it already meets the tolerance.

family	arm	rate	field error	cost
advection	water-filling	$0.0625$ bpv	$17.62\%$	$\mathbf{19.48\pm 0.76\times}$
advection	best low-pass	$0.0625$ bpv	$16.38\%$	$20.11\pm 0.34\times$
advection	SZ3, whole ensemble	$0.0635$ bpv	$151.42\%$	$95.37\pm 1.54\times$
advection	SZ3, per sample, at its floor	$0.8984$ bpv	$100.00\%$	$116.78\pm 0.01\times$
advection	ZFP, per sample, at its floor	$0.3750$ bpv	$100.00\%$	$116.78\pm 0.01\times$
reaction–diffusion	best low-pass	$0.0625$ bpv	$10.97\%$	$\mathbf{10.99\pm 0.87\times}$

Appendix

Table 16: A budget of

64

bits per sample, spent five ways, against the shipping SZ3 and ZFP. Rates are totals including side information and are what the codec actually produced, not what it was asked for; the two per-sample rows could not be brought down to the budget at any error bound and sit at their floors, at

14.4

and

6.0

times the storage the other rows use. Two seeds, full-precision errors

0.857\%

and

0.097\%

. A field error above

100\%

means the reconstruction is further from the data than a field of zeros.

Figure 8: Rate against distortion over seven doublings, four input ensembles. Curves stop at each per-sample arm’s floor, marked by the dotted vertical lines, because no error bound takes it lower. On Darcy real SZ3 becomes exactly lossless at

0.404

bits per value, drawn as a star at the axis floor. Burgers is not a separate panel: PDEBench stores it on the same initial conditions as advection, this comparison compresses inputs only, and the two agree to twelve decimal places.

Figure 9: Every point on a curve costs the same number of bytes. Moving right stores more fields at lower precision, moving left stores fewer at higher. The open ring marks the best point on each line and the dashed line is a full-precision model trained on the whole pool, the configuration the budget would buy if storage were free. Two seeds, error bars are their spread.

family	budget	best point	its error	whole pool	gain
advection	$0.25$ bpv	$250$ at $7.73$ bpv	$\mathbf{1.192\%}$	$2.454\%$	$\mathbf{2.06\times}$
advection	$1.0$ bpv	$4000$ at $1.98$ bpv	$\mathbf{1.080\%}$	$1.318\%$	$1.22\times$
reaction–diffusion	$0.25$ bpv	$4000$ at $0.48$ bpv	$0.0987\%$	$0.1010\%$	$1.02\times$
reaction–diffusion	$1.0$ bpv	$8000$ at $0.99$ bpv	$0.0981\%$	$0.0981\%$	$1.00\times$

Appendix

Table 17: The best point on each budget line against the point that keeps every sample, which is the configuration the rest of this paper uses. Two seeds. Advection at a quarter of a bit is the case that matters: the configuration we have been reporting is the worst of the 8 points on its own budget line, and cutting the sample count thirty-two fold while spending the bits on precision halves the error.

Figure 10: Advection. Left, what the codec stores; right, what the operator pays. Three quantisers: the legacy unconverged Lloyd table, and the converged companded codebook at a per-coefficient cap of

12

and of

14

bits. Lines are probe predictions, hollow circles are trained measurements whose edge colour matches their curve, and the dashed line is the

10\%

acceptability threshold. Both reported costs,

2.85\times

and

1.27\times

, are measured by training.

bar	required	would have failed at	outcome
geometric mean, out of sample	$\leq 1.30\times$	any worse	MISSED , $1.43\times$
probe inside SZ3	$\leq 2.0\times$	any worse	MISSED , $4.65\times$
probe inside a wavelet, worst	$\leq 1.61\times$	any worse	MISSED , $3.49\times$
probe inside a wavelet, median	$\leq 1.25\times$	any worse	MISSED , $2.14\times$
2D Fourier operator under-predicts	$\geq 3$ of $4$	$\leq 2$ of $4$	MISSED , $0$ of $4$
worst case, sixth family	$\leq 2.10\times$	any worse	MISSED , $2.11\times$

Appendix

Table 18: Every bar written into a run script before the run that scored it, from the geometric-mean bar of Appendix C.1 onward. Fifteen of the thirty-eight were missed and twenty were met. The remaining three are marked void rather than counted: they could not have failed, for reasons we did not see when we set them. The two attenuation rows are scored on the single-protocol points; on the earlier two-quantiser assembly they read

2.71

and

1.15

against

1.74

, with the same verdicts.

predictor	in-sample (8)	out-of-sample (8)	mean $\|\log\|$ , out of sample
forward-pass probe	$1.49\times$	$1.61\times$	$0.164$
$15$ -epoch fine-tune	$1.36\times$	$1.66\times$	$0.165$
geometric mean of the two	$1.14\times$	$\mathbf{1.43\times}$	$0.141$

Appendix

Table 19: Worst-case factor off. The pre-registered bar was

1.30\times

out of sample. It was not met.

Figure 11: Left: the geometric-mean predictor on all

16

points, in-sample filled, out-of-sample open, the 2D Fourier neural operator cells ringed. Right: worst-case error per tier, with the pre-registered bar.

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Jun 16, 2026cs.LG

Operator Boosting Produces Pareto-Efficient PDE Surrogates

Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

Lennon J. Shikhman

Georgia Institute of Technology College of Computing Atlanta, Georgia, USA

Neural Operators Neural Partial Differential Equation Solvers Surrogate Models Boosting

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