Variational Physics-informed Neural Operator (VINO) for Solving Partial Differential Equations

• Neural operators learn mappings between function spaces [4] [20], mapping input functions such as initial and boundary conditions to solution functions [22] [23] [24] [25].They are described as universal approximators for continuous operators.Source: §1 · p. 2
• Neural operators struggle to generalize from coarse-resolution training data and to new scenarios or conditions beyond their training data.Source: §1 · p. 2
• Current physics-informed neural operators may require paired input-output data for sufficient accuracy, leaving them dependent on numerical or observational data.Source: §1 · p. 2
• The authors attribute physics-only learning difficulties to strong-form equations requiring higher-order derivatives at collocation points and costly derivative or integral evaluations.Source: §1 · p. 2
• VINO incorporates PDE energy formulations, discretizing the domain and using shape functions to approximate solutions while analytically resolving differentiation and integration.Source: §1 · p. 2
• VINO performs better as mesh size increases and converges on finer meshes, unlike PINO.The introduction also says VINO learns operators purely from physical data with consistent hyperparameters.Source: §1 · p. 2

2 Results

• VINO learns PDE solution operators without paired datasets by minimizing a discretized PDE functional, enabling analytic elementwise integration and differentiation.This avoids reliance on limiting numerical methods or computationally intensive automatic differentiation.Source: §2 · p. 3
• The study compares VINO with FNO and PINO on three benchmark examples, demonstrates solving more complicated cases across three further examples, and summarizes benchmark performance in Table 1 and Figure 2a.It also investigates mesh-grid size effects on accuracy for methods solving the Darcy flow equation.Source: §2 · p. 3
• FNO’s median error increases modestly with mesh size, while its overall error variability remains relatively stable.The passage describes FNO as managing finer meshes effectively.Source: §2 · p. 3
• VINO’s median relative L2 error consistently decreases on finer meshes, with diminishing variability and converging behavior under constant hyperparameters.The mesh-size comparison is shown in Figure 2b.Source: §2 · p. 3

2.1 Comparative Numerical Results

• Across three benchmarks, VINO+data generally has lower relative L2 error; FNO and PINO comparisons were implemented in PyTorch for fairness.The study presents numerical comparisons with FNO and PINO, which were officially released using PyTorch frameworks.Source: §2.1 · p. 3, Figure 2 · p. 5

2.1.1 Second-order anti-derivative

• For the anti-derivative ODE, VINO shows smaller errors on GRF inputs and generalizes better than PINO to sinusoidal and exponential inputs.VINO and data-enhanced VINO have lower mean errors and smaller deviations and variances than PINO, with approximately the same training time.Source: §2.1.1 · p. 7, Figure 3 · p. 8

2.1.2 Poisson Equation

• 0.00751 and 0.00631 are VINO’s standalone and data-enhanced mean L2 errors on Poisson, respectively; data-enhanced VINO’s standard deviation is 0.0026.VINO with data also has lower mean error than PINO with data, and PINO with data does not surpass standalone VINO; training times and computational demands are similar across methods.Source: Table 1 · p. 4, §2.1.2 · pp. 9–10

2.1.3 Darcy Flow

• On Darcy flow, VINO has the lowest mean error, while PINO has the highest mean error and largest standard deviation.The comparison concerns two-dimensional subsurface flow through a porous medium with heterogeneous permeability fields.Source: Table 1 · p. 4, §2.1.3 · p. 11
• Training time across methods is relatively consistent, with small variations.Source: §2.1.3 · p. 11
• Figure 2a's relative L2-error box plots show PINO has the widest spread, while VINO and VINO+data have narrower boxes indicating lower, more consistent errors.Source: §2.1.3 · p. 11
• VINO without data-driven training remains more accurate than PINO even when PINO incorporates data, while adding data reduces mean error and standard deviation for both methods.Source: §2.1.3 · p. 11

2.2.1 Porous Material Structure

• The porous-beam mean error is 0.035 (standard deviation 0.087) on training data and 0.032 (0.051) on test data.VINO's predicted displacement field agrees with the ground-truth solution in the reported train and test samples.Source: §2.2.1 · pp. 12, 14, Figure 6 · p. 12
• After 376 s of training, VINO solves a porous-beam instance in approximately 3 milliseconds, versus 1.5 seconds for IGA.The trained operator maps arbitrary traction and elasticity-modulus distributions to displacement fields.Source: §2.2.1 · pp. 11, 13

2.2.2 Hyperelasticity

• For hyperelasticity, training and testing losses decline steeply before stabilizing, and VINO's test displacement prediction agrees with the ground truth.The loss curves are reported across 1000 iterations; Figure 7 also presents relative L2-error distributions for train and test data.Source: Figure 7 · p. 15, §2.2.2 · p. 16
• After 1282 s of training, VINO solves the problem for any arbitrary function in approximately 0.07 s, compared with 40 seconds on average once training is complete.For a test-dataset sample, the predicted displacement field has very good agreement with the ground-truth solution.Source: §2.2.2 · p. 16

2.2.3 Plate with arbitrary voids

• The plate example formulates a parametric PDE over domains determined by design parameters p drawn from a distribution.The stated goal involves Dirichlet and Neumann boundary conditions; the supplied passage does not report results for this example.Source: §2.2.3 · pp. 16–17
• The plate domain is [0, 5]^2 with arbitrary holes in [1, 4]^2, a clamped left edge, and constant tension traction on the right.Dirichlet and Neumann boundary conditions are fixed, and the domain is represented by uniform meshes.Source: §2.2.3 · p. 17
• The experiment used 1000 training and 200 test samples, with IGA ground truths reserved for comparison rather than training.Displacements were computed at nx = ny = 200 uniformly spaced points and optimized using Adam.Source: §2.2.3 · p. 17
• Relative L2 errors are summarized for training and test data, while losses drop sharply before stabilizing over 1000 iterations.The error box plots show medians and interquartile ranges; loss curves include both training and testing values.Source: §2.2.3 · p. 17
• VINO’s predicted displacement fields show acceptable agreement with the ground-truth solutions for plate samples.Figure 8 presents predictions for training and test samples.Source: §2.2.3 · p. 17

3 Discussion

• VINO incorporates variational physical laws by discretizing the domain and using shape functions to handle derivatives and integrals analytically, enabling learning from physical principles alone.This is presented as an alternative to data-driven methods that rely on potentially costly or unavailable paired input-output datasets.Source: §3 · p. 17
• VINO outperforms PINO, particularly at higher mesh resolutions, where it converges as other methods lose accuracy or perform inconsistently.The authors describe this behavior as evidence of VINO’s robustness and scalability.Source: §3 · p. 17
• VINO currently requires an available variational form; future work should extend it to weak forms, complex multiscale and nonlinear problems, adaptive meshing, and domains that challenge traditional solvers.The study implemented rectangular elements; triangular and tetrahedral elements are suggested as possible adaptations.Source: §3 · p. 17
• For the plate with arbitrary voids, train mean relative L2 error was 0.035 (standard deviation 0.087), and test mean error was 0.032 (standard deviation 0.051).Figure 8 also shows both loss curves rapidly decreasing before stabilizing.Source: Figure 8 · p. 18

4.1 Problem setting

• The problem setting treats PDE parameters as inputs and the unknown solution as output, defining a solution operator that maps each parameter instance to its solution.The parameters can include coefficient or input functions, and boundary conditions may be fixed or included as parameters.Source: §4.1 · p. 19

4.1.1 Solving equation in strong form using PINNs

• Strong-form PINNs approximate a solution with a neural network and minimize a PDE-based loss using exact derivatives computed by automatic differentiation.The PINN loss is defined in the squared norm of the differential equation’s left-hand side.Source: §4.1.1 · p. 19
• Strong-form PINNs can struggle with multiscale or high-frequency solutions and nonlinear hyperbolic PDEs, and require retraining when parameters change.The authors motivate combining neural operators and PINNs in a variational format to improve efficiency.Source: §4.1.1 · pp. 19–20

4.1.2 Solving the variational form using the physics-informed neural networks

• VPINNs minimize a functional rather than enforcing equations at collocation points, and this functional may take negative values unlike the nonnegative squared-error PINN loss.In physical problems, the variational loss has physical dimensions and meaning rather than representing a dimensionless error.Source: §4.1.2 · p. 20
• VPINNs need fewer hyperparameters and can improve computational efficiency and accuracy through lower derivative order, but depend on accurate integration and require a variational form.The Neumann boundary condition is incorporated into the physical loss, avoiding tuning parameters for it.Source: §4.1.2 · pp. 20–21

4.1.3 Learning the Solution Operator Via Neural Operators

• Neural operators approximate the solution operator using stacked layers that combine linear integral operators with pointwise nonlinear activations.The architecture lifts inputs to a higher-dimensional space and projects outputs back to a lower-dimensional space.Source: §4.1.3 · p. 21
• Neural operators rely on paired data and can struggle to generalize across coefficients or geometries; VINO is introduced to address these limitations and overcome data dependence.The passages also note that representative data can be difficult to gather for complex PDEs with slow or unavailable solvers.Source: §4.1.3 · pp. 21–22

4.2 Learning the Solution Operator Via Variational Physics-Informed Neural Operator (VINO)

• VINO minimizes the PDE variational functional across training inputs without paired datasets, using domain discretization inspired by the finite element method to improve robustness to resolution changes.The functional is used directly as the loss for the neural operator.Source: §4.2 · p. 22
• VINO divides the domain into elements and evaluates derivatives and integrals analytically, avoiding fine-grid finite-difference costs and potential numerical instabilities.The rectangular four-node element illustrates the approach; its shape functions have linear behavior along element edges to ensure interelement C0 continuity.Source: §4.2 · pp. 22–23
• For Poisson’s equation, VINO minimizes a discretized energy functional whose domain value is the sum of element contributions, without finite-element-style assembly.The same process can be used for other PDEs and element types.Source: §4.2 · p. 24
• For the Poisson equation on a 16×16 grid, relative L2 errors were 0.1377 (Midpoint), 0.1378 (Trapezoidal), 0.1583 (Simpson), and 0.0181 (VINO), showing VINO’s lower error.Source: §4.2 · p. 24, Figure 9 · p. 25

• The authors declare no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Source: Declaration of Competing Interest · p. 30