stat.MLOct 13, 2026

Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations

Authors: Wonjae Lee, Jiuk Jang, Hyungbin Park

Organizations: InnoCore AI Meta-Scientist Research Group, KAIST, Daejeon 34141, South Korea. · Department of Mathematical Sciences, Seoul National University, Seoul 08826, South Korea. · Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea.

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Abstract

This paper studies Wong--Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong--Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.

Figures & tables

TaskKernel K(t)K(t)Drift bbHHϵ\epsilonbb parameter
T1\mathcal{T}_{1}exp⁡(−2t)\exp(-2t)b(y)=yb(y)=y––1.01.0
T2\mathcal{T}_{2}(t+ϵ)H−1/2(t+\epsilon)^{H-1/2}b(y)=byb(y)=by0.400.401\times10−41\text{\times}{10}^{-4}0.080.08
T3\mathcal{T}_{3}(t+ϵ)H−1/2(t+\epsilon)^{H-1/2}b(y)=bb(y)=b0.400.401\times10−41\text{\times}{10}^{-4}0.080.08
Table 1: Volterra tasks.
ModelArchitectureLRStepLR (s,γ)(s,\gamma)Val. MSE
MFNOm=64, w=64, d=6, k=15×10−45\times 10^{-4}(25,0.9)(25,0.9)2.44151×10−42.44151\times 10^{-4}
ZFNOm=64, w=64, d=6, k=11×10−31\times 10^{-3}(100,0.7)(100,0.7)1.28829×10−41.28829\times 10^{-4}
FNOm=64, w=64, d=6, k=15×10−45\times 10^{-4}(100,0.7)(100,0.7)3.69936×10−43.69936\times 10^{-4}
LSTMh=128, layers=1, drop=0.01×10−31\times 10^{-3}(25,0.9)(25,0.9)9.78445×10−49.78445\times 10^{-4}
TCNw=512, blocks=6, k=2, drop=0.21×10−31\times 10^{-3}(25,0.9)(25,0.9)4.50948×10−44.50948\times 10^{-4}
DeepONetp=300, branch=3, trunk=2, w=1282×10−42\times 10^{-4}(25,0.9)(25,0.9)1.52127×10−21.52127\times 10^{-2}
Table 2: Selected hyperparameters for T1\mathcal{T}_{1} .
ModelArchitectureLRStepLR (s,γ)(s,\gamma)Val. MSE
MFNOm=64, w=64, d=6, k=12×10−42\times 10^{-4}(50,0.7)(50,0.7)7.59577×10−47.59577\times 10^{-4}
ZFNOm=64, w=64, d=6, k=12×10−42\times 10^{-4}(50,0.7)(50,0.7)4.67849×10−44.67849\times 10^{-4}
FNOm=32, w=64, d=6, k=12×10−42\times 10^{-4}(25,0.9)(25,0.9)8.71486×10−48.71486\times 10^{-4}
LSTMh=128, layers=1, drop=0.01×10−31\times 10^{-3}(50,0.9)(50,0.9)2.07717×10−32.07717\times 10^{-3}
TCNw=512, blocks=6, k=2, drop=0.21×10−31\times 10^{-3}(100,0.7)(100,0.7)1.86754×10−31.86754\times 10^{-3}
DeepONetp=384, branch=4, trunk=3, w=2562×10−42\times 10^{-4}(50,0.7)(50,0.7)6.95737×10−26.95737\times 10^{-2}
Table 3: Selected hyperparameters for T2\mathcal{T}_{2} .
ModelArchitectureLRStepLR (s,γ)(s,\gamma)Val. MSE
MFNOm=64, w=64, d=6, k=12×10−42\times 10^{-4}(50,0.7)(50,0.7)5.42639×10−45.42639\times 10^{-4}
ZFNOm=64, w=64, d=6, k=12×10−42\times 10^{-4}(100,0.9)(100,0.9)4.81996×10−44.81996\times 10^{-4}
FNOm=32, w=32, d=6, k=12×10−42\times 10^{-4}(25,0.9)(25,0.9)7.18237×10−47.18237\times 10^{-4}
LSTMh=128, layers=1, drop=0.01×10−31\times 10^{-3}(50,0.9)(50,0.9)2.60186×10−32.60186\times 10^{-3}
TCNw=128, blocks=6, k=3, drop=0.21×10−31\times 10^{-3}(25,0.9)(25,0.9)1.47409×10−31.47409\times 10^{-3}
DeepONetp=300, branch=3, trunk=2, w=1282×10−42\times 10^{-4}(50,0.7)(50,0.7)6.71098×10−26.71098\times 10^{-2}
Table 4: Selected hyperparameters for T3\mathcal{T}_{3} .
Figure 1: Training loss comparison for T1\mathcal{T}_{1} , seed 1.
ModelT1\mathcal{T}_{1}T2\mathcal{T}_{2}T3\mathcal{T}_{3}
MFNO1.451×10−21.451\times 10^{-2} ±\pm 4.150×10−44.150\times 10^{-4}2.020×10−22.020\times 10^{-2} ±\pm 5.575×10−45.575\times 10^{-4}1.978×10−21.978\times 10^{-2} ±\pm 4.828×10−44.828\times 10^{-4}
ZFNO1.050×10−2\boldsymbol{1.050\times 10^{-2}} ±\pm 5.232×10−45.232\times 10^{-4}1.711×10−2\boldsymbol{1.711\times 10^{-2}} ±\pm 2.779×10−42.779\times 10^{-4}1.855×10−2\boldsymbol{1.855\times 10^{-2}} ±\pm 4.354×10−44.354\times 10^{-4}
FNO1.720×10−21.720\times 10^{-2} ±\pm 1.153×10−31.153\times 10^{-3}2.363×10−22.363\times 10^{-2} ±\pm 1.329×10−31.329\times 10^{-3}2.330×10−22.330\times 10^{-2} ±\pm 6.305×10−46.305\times 10^{-4}
LSTM2.827×10−22.827\times 10^{-2} ±\pm 1.932×10−31.932\times 10^{-3}3.797×10−23.797\times 10^{-2} ±\pm 3.254×10−33.254\times 10^{-3}4.505×10−24.505\times 10^{-2} ±\pm 2.773×10−32.773\times 10^{-3}
TCN2.111×10−22.111\times 10^{-2} ±\pm 2.770×10−32.770\times 10^{-3}4.873×10−24.873\times 10^{-2} ±\pm 1.529×10−21.529\times 10^{-2}3.167×10−23.167\times 10^{-2} ±\pm 4.304×10−34.304\times 10^{-3}
DeepONet1.026×10−11.026\times 10^{-1} ±\pm 3.474×10−33.474\times 10^{-3}1.664×10−11.664\times 10^{-1} ±\pm 3.852×10−33.852\times 10^{-3}1.818×10−11.818\times 10^{-1} ±\pm 6.010×10−36.010\times 10^{-3}
Table 5: Resolution- 128128 relative ℓ2\ell_{2} errors. TCN produced no NaNs on T1\mathcal{T}_{1} , 5 NaNs out of 10 seeds on T2\mathcal{T}_{2} , and 8 NaNs out of 10 seeds on T3\mathcal{T}_{3} ; the TCN statistics for T2\mathcal{T}_{2} and T3\mathcal{T}_{3} use only the finite seeds.
ModelT1\mathcal{T}_{1}T2\mathcal{T}_{2}T3\mathcal{T}_{3}
MFNO5.009×10−25.009\times 10^{-2} ±\pm 1.756×10−31.756\times 10^{-3}3.2×10−23.2\times 10^{-2} ±\pm 4.899×10−44.899\times 10^{-4}3.472×10−23.472\times 10^{-2} ±\pm 7.024×10−47.024\times 10^{-4}
ZFNO3.234×10−2\boldsymbol{3.234\times 10^{-2}} ±\pm 1.33×10−31.33\times 10^{-3}2.676×10−2\boldsymbol{2.676\times 10^{-2}} ±\pm 4.723×10−44.723\times 10^{-4}3.107×10−2\boldsymbol{3.107\times 10^{-2}} ±\pm 5.798×10−45.798\times 10^{-4}
FNO4.942×10−24.942\times 10^{-2} ±\pm 1.473×10−31.473\times 10^{-3}3.943×10−23.943\times 10^{-2} ±\pm 1.531×10−31.531\times 10^{-3}5.144×10−25.144\times 10^{-2} ±\pm 1.864×10−31.864\times 10^{-3}
LSTM1.024×10−11.024\times 10^{-1} ±\pm 4.219×10−34.219\times 10^{-3}9.632×10−29.632\times 10^{-2} ±\pm 3.493×10−33.493\times 10^{-3}1.021×10−11.021\times 10^{-1} ±\pm 4.259×10−34.259\times 10^{-3}
TCN4.756×10−24.756\times 10^{-2} ±\pm 4.487×10−34.487\times 10^{-3}8.505×10−28.505\times 10^{-2} ±\pm 1.65×10−21.65\times 10^{-2}4.91×10−24.91\times 10^{-2} ±\pm 2.328×10−32.328\times 10^{-3}
DeepONet2.045×10−12.045\times 10^{-1} ±\pm 5.952×10−35.952\times 10^{-3}2.433×10−12.433\times 10^{-1} ±\pm 5.031×10−35.031\times 10^{-3}2.783×10−12.783\times 10^{-1} ±\pm 7.847×10−37.847\times 10^{-3}
Table 6: Resolution- 128128 relative ℓ∞\ell_{\infty} errors. TCN produced no NaNs on T1\mathcal{T}_{1} , 5 NaNs out of 10 seeds on T2\mathcal{T}_{2} , and 8 NaNs out of 10 seeds on T3\mathcal{T}_{3} ; the TCN statistics for T2\mathcal{T}_{2} and T3\mathcal{T}_{3} use only the finite seeds.
Figure 2: Resolution generalization for the FNO-family models. Error bars show sample standard deviation over seeds.
ModelParametersAvg. time (s)Std. time (s)ms/pathSeeds
MFNO1 606 3371\,606\,3372.866\times10−22.866\text{\times}{10}^{-2}4.373\times10−34.373\text{\times}{10}^{-3}1.399\times10−21.399\text{\times}{10}^{-2}10
ZFNO1 606 3371\,606\,3372.625\times10−22.625\text{\times}{10}^{-2}1.208\times10−41.208\text{\times}{10}^{-4}1.282\times10−21.282\text{\times}{10}^{-2}10
FNO1 606 3371\,606\,3371.926\times10−21.926\text{\times}{10}^{-2}8.682\times10−48.682\text{\times}{10}^{-4}9.402\times10−39.402\text{\times}{10}^{-3}10
LSTM67 71367\,7139.902\times10−39.902\text{\times}{10}^{-3}2.834\times10−32.834\text{\times}{10}^{-3}4.835\times10−34.835\text{\times}{10}^{-3}10
TCN5 784 6045\,784\,6041.879\times10−11.879\text{\times}{10}^{-1}1.606\times10−41.606\text{\times}{10}^{-4}9.173\times10−29.173\text{\times}{10}^{-2}10
DeepONet160 344160\,3448.289\times10−38.289\text{\times}{10}^{-3}2.719\times10−32.719\text{\times}{10}^{-3}4.047\times10−34.047\text{\times}{10}^{-3}10
Table 7: T1\mathcal{T}_{1} parameter counts and inference timing.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

ModelCandidate 1Candidate 2Candidate 3Candidate 4Candidate 5
MFNO(16,16,4,1)(16,16,4,1)(32,32,4,1)(32,32,4,1)(32,32,5,1)(32,32,5,1)(32,64,5,1)(32,64,5,1)(64,64,6,1)(64,64,6,1)
ZFNO(16,16,4,1)(16,16,4,1)(32,32,4,1)(32,32,4,1)(32,32,6,1)(32,32,6,1)(32,64,6,1)(32,64,6,1)(64,64,6,1)(64,64,6,1)
FNO(16,16,4,1)(16,16,4,1)(32,32,4,1)(32,32,4,1)(32,32,6,1)(32,32,6,1)(32,64,6,1)(32,64,6,1)(64,64,6,1)(64,64,6,1)
Appendix
Table 8: Architecture candidates for the FNO-family models.
ModelCandidate 1Candidate 2Candidate 3Candidate 4Candidate 5
TCN(128,3,2,0.1)(128,3,2,0.1)(128,6,3,0.2)(128,6,3,0.2)(256,4,2,0.2)(256,4,2,0.2)(256,6,2,0.2)(256,6,2,0.2)(512,6,2,0.2)(512,6,2,0.2)
LSTM(128,1,0.0)(128,1,0.0)(128,2,0.0)(128,2,0.0)(256,2,0.1)(256,2,0.1)(512,2,0.2)(512,2,0.2)(512,3,0.2)(512,3,0.2)
DeepONet(128,2,2,128)(128,2,2,128)(128,3,2,128)(128,3,2,128)(300,3,2,128)(300,3,2,128)(384,4,3,128)(384,4,3,128)(384,4,3,256)(384,4,3,256)
Appendix
Table 9: Architecture candidates for TCN, LSTM, and DeepONet.
TaskModelResolutionℓ2\ell_{2} mean ±\pm stdℓ∞\ell_{\infty} mean ±\pm std
T1\mathcal{T}_{1}FNO1601.729\times10−021.729\text{\times}{10}^{-02} ±\pm 1.215\times10−031.215\text{\times}{10}^{-03}5.148\times10−025.148\text{\times}{10}^{-02} ±\pm 1.483\times10−031.483\text{\times}{10}^{-03}
T1\mathcal{T}_{1}FNO1921.678\times10−021.678\text{\times}{10}^{-02} ±\pm 1.242\times10−031.242\text{\times}{10}^{-03}5.128\times10−025.128\text{\times}{10}^{-02} ±\pm 1.337\times10−031.337\text{\times}{10}^{-03}
T1\mathcal{T}_{1}FNO2561.702\times10−021.702\text{\times}{10}^{-02} ±\pm 1.242\times10−031.242\text{\times}{10}^{-03}5.328\times10−025.328\text{\times}{10}^{-02} ±\pm 1.327\times10−031.327\text{\times}{10}^{-03}
T1\mathcal{T}_{1}FNO3201.674\times10−021.674\text{\times}{10}^{-02} ±\pm 1.309\times10−031.309\text{\times}{10}^{-03}5.405\times10−025.405\text{\times}{10}^{-02} ±\pm 1.321\times10−031.321\text{\times}{10}^{-03}
T1\mathcal{T}_{1}FNO3841.773\times10−021.773\text{\times}{10}^{-02} ±\pm 1.334\times10−031.334\text{\times}{10}^{-03}5.633\times10−025.633\text{\times}{10}^{-02} ±\pm 1.400\times10−031.400\text{\times}{10}^{-03}
T1\mathcal{T}_{1}FNO5121.690\times10−021.690\text{\times}{10}^{-02} ±\pm 1.321\times10−031.321\text{\times}{10}^{-03}5.610\times10−025.610\text{\times}{10}^{-02} ±\pm 1.332\times10−031.332\text{\times}{10}^{-03}
Appendix
Table 10: Full resolution-generalization errors for MFNO, ZFNO, and FNO.
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