Adapting neural operators for mechanics decisions under changing operating conditions

Prashant K. Jha; Koffi Enakoutsa; Ian Galloway; Henry Anderson

cs.CESep 27, 2026

Adapting neural operators for mechanics decisions under changing operating conditions

Authors: Prashant K. Jha, Koffi Enakoutsa, Ian Galloway, Henry Anderson

Organizations: Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA · Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA

Abstract

Neural operators can accelerate repeated nonlinear mechanics calculations, but their accuracy can deteriorate as operating conditions move beyond the training range. This work studies whether high-fidelity solutions acquired during use can be reused to adapt a neural operator and improve subsequent mechanics-based command selection. Two hard-magnetic soft-material systems are simulated using high-fidelity finite-element (FE) models, providing reference solutions for evaluating surrogate predictions and selected commands. A neural operator predicts deformation from known material, loading, and magnetic-field inputs, while an empirical error estimator determines which predictions may be used for command selection. Selected FE evaluations supplement these predictions, and their complete loading paths are retained for periodic updates of the neural operator and estimator. In both examples, the fixed operator loses substantial accuracy when stiffness and loading move outside the training range. Updates using 16 acquired paths recover much of the lost accuracy while preserving accuracy in the nominal regime. Under the same FE evaluation budget, the updated operators also improve command selection, although the benefit varies with the operating condition. Error estimation is less consistent, with inaccurate predictions sometimes accepted and accurate predictions rejected. These results demonstrate that reusing high-fidelity loading paths can extend the useful operating range of a neural operator. However, improved forward accuracy alone does not guarantee reliable prediction acceptance, highlighting prediction-specific error assessment as a separate requirement for trustworthy decision making.

Figures & tables

Figure 1: The two hard-magnetic soft-material structures considered in this study. (a) A compliant positioning platform supported by four magnetically active ligaments and clamped at the outer roots. (b) A gripping wishbone with two magnetically active fingers supported by passive lower branches and clamped along the base. Arrows within the active regions show remanent magnetization, and grey denotes passive material. Dark outlines show representative FE deformations; the platform displacement is magnified twofold, while the wishbone is shown at true scale.

Panel of Figure 2 — Figure 2: Reference and surrogate-assisted command-selection sequence. For a fixed operating condition $\bm{\theta}$ , each candidate command $\bm{a}\in\mathcal{A}$ defines the known forward input $\bm{m}=(\bm{a},\bm{\theta})$ . The neural operator predicts the displacement and associated mechanics QoIs, while selected commands are evaluated using the high-fidelity FE model. The QoI-error estimator determines which neural-operator predictions may enter the command calculation. The available set comprises commands with accepted neural-operator predictions and the FE-evaluated commands. An admissible command is selected from this set by minimizing the loss. Acquired FE paths are retained for periodic updates of the neural operator and estimator.

Figure 3: Platform quantities of interest. The translations

d_{x}

and

d_{y}

are area-averaged displacements of the stiff platform

\Omega_{P}

, and

\phi_{P}

is its finite rigid-body rotation. The displaced outline is exaggerated to illustrate the pose quantities. The yellow regions mark the four ligament roots and four ligament–platform junctions;

q_{99}

is the maximum of their area-weighted 99th-percentile equivalent strains.

Figure 4: Wishbone quantities of interest. The points

\bm{c}_{L}

and

\bm{c}_{R}

represent averages of the deformed positions along the highlighted inner tip segments, weighted by reference arclength. Their horizontal separation defines the aperture

g

, the horizontal coordinate of their midpoint defines

c_{x}

, and their signed vertical difference defines

\delta_{y}

. The shaded root region

\Omega_{W}^{q}

is used to calculate the area-weighted 99th-percentile equivalent strain

q_{99}

. The reference configuration is shown in light grey, and the deformed configuration illustrates the QoI definitions.

Nominal FE data
Quantity	Specification
Training paths	200
Validation paths	25
Held-out test paths	30
Retained increments per training path	16
Coefficient network

Table 1: Data, coefficient-network architecture, and initial training settings for each example.

Figure 5: Platform geometry and representative FE responses. (a) Reference geometry, mesh, clamped roots, and loaded platform edge. The yellow and cyan regions identify the root and junction regions used to calculate $q_{99}$ . (b) Platform translation and rotation. (c)–(e) FE deformations at $T_{1}$ in Table 3 , for $B=0$ and for $B=$80\text{\,}\mathrm{m}\mathrm{T}$$ at $\alpha=-20^{\circ}$ and $+20^{\circ}$ , respectively. Displacements in the deformation panels are amplified by a factor of two. — Figure 5: Platform geometry and representative FE responses. (a) Reference geometry, mesh, clamped roots, and loaded platform edge. The yellow and cyan regions identify the root and junction regions used to calculate $q_{99}$ . (b) Platform translation and rotation. (c)–(e) FE deformations at $T_{1}$ in Table 3 , for $B=0$ and for $B=$ 80\text{,}\mathrm{m}\mathrm{T}$$ at $\alpha=-20^{\circ}$ and $+20^{\circ}$ , respectively. Displacements in the deformation panels are amplified by a factor of two.

Input range	$\gamma_{G}$	$f_{x}^{P}$ (mN)	Preliminary median $E_{Q}$
Nominal	$[0.95,1.05]$	$[-0.10,0.10]$	—
S1	$[0.86,0.94]$	$[0.25,0.70]$	0.275
S2	$[0.78,0.89]$	$[0.45,1.00]$	0.735
S3	$[0.72,0.84]$	$[0.70,1.30]$	1.181

Table 2: Nominal and shifted input ranges for the platform. The magnetic-field ranges are those in Eq. 41 , and

f_{y}^{P}\in[-0.65,0]\

\mathrm{m}\mathrm{N}$$ for every set. Each shifted range is assessed using 12 preliminary paths.

Adaptation				Command tests
Condition	$\gamma_{G}$	$f_{x}^{P}$	$f_{y}^{P}$	Condition	$\gamma_{G}$	$f_{x}^{P}$	$f_{y}^{P}$
$A_{1}$	0.816	0.82	-0.4875	$T_{1}$	0.804	0.88	-0.260
$A_{2}$	0.792	0.94	-0.1625	$T_{2}$	0.780	1.00	-0.520
$A_{3}$	0.768	1.06	-0.3250	$T_{3}$	0.756	1.12	-0.130
$A_{4}$	0.744	1.18	-0.4875	$T_{4}$	0.732	1.24	-0.390

Table 3: Platform operating conditions. Force components are in mN. Four FE paths are acquired at each

A_{k}

. The conditions

T_{1}

through

T_{4}

are reserved for testing command selection.

Figure 6: Platform FE solution and initial neural-operator prediction for the 15th of 30 nominal test paths ordered by the initial operator’s $E_{Q}$ . The first two panels use the same displacement-magnitude scale; the third shows the pointwise displacement error. All fields are shown on the reference configuration.

Figure 7: Platform FE solution, initial prediction, and accepted 16-path updated prediction for the same S3 test input. The input is the 15th of 30 paths ordered by the initial operator’s $E_{Q}$ . The update uses random-acquisition seed 1. Displacement panels share one color scale, and error panels share a separate scale. Results across all five acquisition sequences are given in Fig. 8 .

Figure 8: Platform QoI errors before and after adaptation. The leftmost distribution uses the 30 nominal test paths; the remaining distributions use the 30 S3 test paths from Table 2 . The five random-acquisition sequences are shown individually. Only estimator-guided seed 3 satisfies the update-acceptance requirements at both stages. The shaded candidate updates reduce shifted error but are not used for command selection because they fail validation.

Figure 9: Initial platform error estimate versus FE-evaluated QoI error on the 30 S3 test paths. Acceptance requires

\widehat{E}_{Q}\leq 1

, while the actual accuracy requirement is

E_{Q}\leq 1

. The estimator accepts one inaccurate prediction and rejects eight accurate predictions. Its rejection of 28 of 30 predictions and negative rank correlation indicate poor discrimination between acceptable and unacceptable errors.

Figure 10: FE responses of all 41 platform commands in the $(d_{x},\phi_{P})$ plane at $T_{1}$ for the eighth target, labeled T8. The dashed ellipses are contours of the pose loss $L_{P}$ about T8. The initial hybrid calculation selects command 29 rather than the FE-optimal command 23. The accepted 16-path random update selects command 23 while using the same four FE evaluations. The remaining FE responses are used only to assess the selected commands.

Figure 11: Platform command results before adaptation and after eight and 16 acquired paths at the conditions in Table 3 . The upper row shows median FE-evaluated regret over eight targets for each acquisition sequence; the lower row shows the fraction of targets selecting the FE-optimal command. Random and estimator-guided acquisition already select different four-command FE sets before adaptation, so their initial differences are not caused by learning. Sequences whose operator updates fail validation retain

M_{0}

, although refitting the estimator and selecting a new four-command FE set can change their decisions.

Figure 12: Unique platform FE loading paths and recorded FE solution time by purpose. The nominal category includes training, validation, and testing. Complete-command reference calculations are used to assess the selected commands; only four selected FE evaluations enter each hybrid command calculation. These totals do not constitute a measurement of the time required for one complete surrogate-assisted decision.

Figure 13: Wishbone geometry, candidate commands, and FE responses. (a) Plane-strain geometry and mesh, clamped base, right-tip force, and active and passive regions. The fingers have

\phi=0.30

and

|\bm{B}^{r}|=

40\text{,}\mathrm{m}\mathrm{T}

, with mirrored remanent directions indicated by arrows. (b) The 21 candidate commands in the $(B_{x},B_{y})$ plane. (c) FE deformations at $T_{2}$ in Table 4 : the zero-field command $a_{0}$ and commands $a_{16}$ and $a_{20}$ with $B=$80\text{\,}\mathrm{m}\mathrm{T}

and

\alpha=-20^{\circ}

and

+20^{\circ}

, respectively. Deformations are shown at true scale over the reference mesh.

Condition	Role	$\gamma_{G}$	$f_{x}^{R}$	$f_{y}^{R}$	$d_{\mathrm{box}}$
$A_{1}$	Adaptation	0.85	1.25	0.80	0.4180
$A_{2}$	Adaptation	0.83	1.40	0.90	0.5659
$A_{3}$	Adaptation	0.80	1.75	1.25	0.8975
$A_{4}$	Adaptation	0.78	2.00	1.50	1.1421
$T_{1}$	Command test	0.8693	1.0011	1.1006	0.3259
$T_{2}$	Command test	0.8086	1.1120	1.1309	0.5757

Table 4: Wishbone operating conditions. Force components are in mN. Four FE paths are acquired at each

A_{k}

; the conditions

T_{1}

through

T_{4}

are reserved for testing command selection. The last column gives the normalized distance outside the nominal input box.

Figure 14: Wishbone FE solution and initial prediction for the 16th of 30 nominal test paths ordered by the initial operator’s $E_{Q}$ . Panels (a) and (b) use a common displacement-magnitude scale; panel (c) shows the pointwise displacement error on a separate scale. Fields are shown on their deformed configurations at true scale over the reference mesh.

Figure 15: Wishbone FE solution, initial prediction, and 16-path updated prediction for the same held-out shifted input. The input is the 16th of 30 paths ordered by the initial operator’s

E_{Q}

, and the illustrated update uses estimator-guided training seed 1. Panels (a), (b), and (e) share one displacement scale; panels (c) and (f) share one error scale. Deformations are shown at true scale. Results across all five seeds are reported in Table 5 .

Operator	Shifted paths	$T_{1}$	$T_{2}$	$T_{3}$	$T_{4}$
$M_{0}$	5.097	1.353	2.299	5.702	11.373
$M_{1}$ , estimator-guided	3.439	0.785	1.192	3.030	7.532
$M_{2}$ , estimator-guided	1.132	1.024	0.770	0.775	1.131
$M_{2}$ , random	0.803	0.946	0.819	0.878	1.149

Table 5: Wishbone normalized QoI errors before and after adaptation. The shifted-path column uses the 30 test paths from Eq. 43 ; each

T_{k}

column uses the 21 candidate commands at that condition in Table 4 . Entries for

M_{0}

are medians over the corresponding inputs. Updated-operator entries are medians across five training seeds of those input-wise medians.

Condition	Hybrid	Four FE	Hybrid	Four FE	Surrogate use
	Agreement		Mean regret
$T_{1}$	5	1	0.045	1.099	5/5; false acceptance
$T_{2}$	3	1	0.503	1.213	4/5; no false acceptance
$T_{3}$	3	2	0.360	1.174	3/5; no false acceptance
$T_{4}$	2	2	1.031	1.031	1/5; one false acceptance

Table 6: Wishbone command selection using estimator-guided

M_{2}

operators at the conditions in Table 4 . Agreement counts FE-optimal selections among eight targets. Regret is averaged over eight targets for each operator. Both statistics are then summarized by the median across five training seeds. The last column gives the number of seeds using additional accepted surrogate predictions and indicates whether false acceptance occurs.

Figure 16: Wishbone forward accuracy and command selection after adaptation. (a) Median

E_{Q}

on the shifted path test and the four conditions in Table 4 . For the updated operators, individual markers show training seeds and the lines give five-seed medians; the

M_{0}

curve represents the single fixed operator. (b) Mean regret over eight targets for the estimator-guided

M_{2}

hybrid and four-FE calculations; open circles show individual seeds, and bars give the median across seeds. Annotations report the number of seeds using additional surrogate predictions and total false acceptances.

Aspect	Platform	Wishbone
Forward adaptation	Shifted median $E_{Q}$ decreases from 1.147 to 0.144	Shifted median $E_{Q}$ decreases from 5.097 to 0.803
Error estimation	Initial estimator rejects most shifted predictions but poorly distinguishes their errors	Some estimators falsely accept predictions after adaptation
Command selection	Zero median regret at all four test conditions after adaptation; individual errors remain	Hybrid improves on the four-FE calculation at $T_{2}$ and $T_{3}$ ; equal median regret at $T_{4}$
Distinct acquired data	Two estimator-guided and five random 16-path sets	One estimator-guided and five random 16-path sets

Table 7: Comparison of the principal findings. Updated forward errors correspond to 16 randomly acquired paths and are summarized by the median of the five per-operator median errors. The examples use different QoIs, tolerances, and shifted operating ranges.

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Neural operators are data-driven models that learn mappings from inputs that parameterize partial differential equations, such as spatially varying coefficients, initial conditions, forcing terms, boundary conditions, or geometries, to solution fields or quantities of interest. Once trained, they can serve as surrogates for classical numerical solvers in many-query settings that require repeated evaluations for varying inputs. We address the question of when, and then why, neural operator surrogates outperform classical numerical solvers, in terms of cost for a given accuracy. We focus on the post-training, many-query limit, in which data-acquisition and training costs are treated as fixed and fully amortized. Even in this deliberately favorable regime for neural operators, there are regimes in which classical solvers outperform the surrogate models. We compare the cost-accuracy performance of neural operator surrogates and classical numerical solvers through a reproducible benchmark study comparing neural operators with problem-matched classical solvers on representative problems in computational science and engineering, focusing on prediction error, per-query floating-point cost, and wall-clock runtime. Neural operators are most competitive at low-to-moderate accuracy requirements. Their floating-point cost advantage depends strongly on the problem structure, arising when they avoid temporal or nonlinear iterations or predict a reduced quantity of interest rather than a full solution field. Additional wall-clock speedups result from dense tensor operations that are well suited to modern hardware. As the target accuracy is tightened, achieving the required accuracy with neural operators becomes increasingly challenging, and classical solvers outperform surrogates in this regime; thus classical solvers will remain important for verification and high-accuracy computation.

Abstract

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