ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

Subhendu Maity; Pratibhamoy Das; Arihant Patawari; Ameya D. Jagtap

Source-linked AI summary

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

Subhendu Maity, Pratibhamoy Das, Arihant Patawari, Ameya D. Jagtap

arXiv:2608.29499v1Aug 30, 2026math.NAmath-ph

TL;DR

PINNs can have difficulty resolving sharp boundary and interior gradients, while existing weak-loss and gradient-enhanced variants do not fully address this problem. The paper introduces ENPINN, combining variational test functions with modified residual-derivative terms and an energy-norm analysis. Across diverse transport benchmarks, ENPINN is reported to improve accuracy and convergence, though its residual derivatives increase computational cost and its scope is currently limited.

Problem
PINNs may poorly resolve sharp boundary and interior gradients, and the theoretical basis for gradient-enhanced loss terms remains limited.
Method
ENPINN combines an energy-norm weak-form loss, variational test functions, and spatial derivatives of weighted residuals, supported by existence, stability, derivative-bound, and error analyses.
Results
Numerical experiments report faster convergence and higher accuracy than gPINN, WLPINN, and VSPINN across combustion, Burgers interior-layer, three-dimensional, and coupled multiscale problems.
Takeaways & Limitations
ENPINN provides a theoretically grounded framework for detecting sharp boundary and interior layers in generalized transport problems.
Takeaways & Limitations
The formulation is primarily designed for generalized convection–diffusion–reaction problems with sharp layers, while moving-layer settings may require substantial extensions and residual derivatives raise computational cost to gPINN-like levels.

Abstract

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Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

1. Introduction

The paper targets PINN difficulty with sharp boundary and interior gradients by combining variational test functions, residual derivatives, and an energy-norm objective. ENPINN is theoretically analyzed and numerically compared with related PINN variants across layer-dominated transport problems.

PINNs can struggle to resolve boundary layers because spectral bias limits their representation of high-frequency and multiscale features.
Generalized convection–diffusion–reaction problems involve multiple perturbation mechanisms that can produce sharp boundary layers and steep gradients.The paper notes that small L2 residual error does not necessarily ensure accurate gradient resolution.
ENPINN combines gradient-based information with a variational loss formulation to improve predictive accuracy in the energy norm while retaining PINN interpretability.
WLPINN improves boundary-condition enforcement through residual scaling, whereas gPINN adds residual derivatives; neither formulation alone adequately captures steep solution gradients.The analysis identifies larger energy-norm errors for WLPINN and insufficiency of interior derivative terms in gPINN.
The framework establishes ENPINN existence, derivative bounds, and energy-error estimates, and extends the architecture to coupled multiscale problems.
Numerical comparisons use the energy norm for convection–diffusion–reaction problems, including combustion, coupled multiscale, Burgers interior-layer, and three-dimensional cases.

2. Mathematical Formulation with Properties of Continuous Solution

The paper formulates a generalized time-dependent convection–diffusion–reaction problem with small diffusion and convection parameters, whose solution may contain boundary and corner layers. Perturbation-dependent structure and derivative bounds characterize the continuous solution.

2.1. Formulation of Governing Problem: The application concerns generalized transport problems derived from heat transfer, chemical kinetics, and diffusion, with nonlinear dependence on one unknown variable.
2.1. Formulation of Governing Problem: The governing model is a time-dependent convection–diffusion–reaction equation with diffusion, convection, reaction, initial, and homogeneous spatial boundary terms.
2.1. Formulation of Governing Problem: Boundary-layer behavior occurs for small ε and µ, and the solution’s qualitative behavior and regularity depend on the ratio µ^2/ε.
2.1. Formulation of Governing Problem: The formulation uses Sobolev spaces W^P,Q(Ω), with H^P(Ω)=W^P,2(Ω) and W^0,0(Ω) equivalent to L2(Ω).
2.2. Nature of the boundary layers: The spatial boundary is divided into left, right, bottom, and top portions to analyze distinct boundary-layer contributions.
2.2. Nature of the boundary layers: The solution is decomposed into a regular component, four boundary-layer components, and four corner-layer functions associated with the domain’s boundaries and corners.
2.2. Nature of the boundary layers: Derivative bounds are established for the solution components under compatibility conditions and the regime µ^2 ≤ ζε/α0.

3. Formulation of ENPINN

ENPINN uses neural-network approximations trained on interior, boundary, and initial points with modified residuals and their spatial derivatives. A boundary-scaling test function, quadrature-based loss, and optimization procedure together target accurate energy-norm resolution of sharp gradients.

3.1. Mathematical Architecture of Neural Networks: The neural network is a feedforward architecture with depth H, layer widths d_h, trainable weights and biases, and an activation function selected for the target output.
3.2. Training Points: Training points are partitioned into interior points for PDE residuals, spatial-boundary points for boundary conditions, and initial points at t=0.
3.3. Modified Residuals: ENPINN defines residuals using the neural-network approximation and includes interior, spatial-boundary, and initial-condition residuals.
3.3. Modified Residuals: The interior residual is multiplied by a test function V(x,y) that becomes small near expected boundary layers, scaling the residual in those regions.
3.3. Modified Residuals: Derivative interior residuals include both derivatives of the PDE residual and terms involving derivatives of the test function.The x- and y-direction terms are N(int,p)x=NresxV+NresVx and N(int,p)y=NresyV+NresVy.
3.4. Quadrature Rules: Quadrature approximates integrals using nodes and positive weights, while ENPINN weights may additionally depend on perturbation parameters such as ε or µ.Monte Carlo quadrature is described as using independent and identically distributed points with a bounded quadrature error.
3.5. Loss Function: The ENPINN loss combines weighted initial, boundary, interior, and interior-residual-derivative contributions using penalty coefficients and quadrature weights.
3.6. Algorithm for ENPINN: The algorithm minimizes the ENPINN loss over network parameters using adaptive learning rates and Adam until an epoch or loss threshold is reached.The loss contains boundary, initial, interior, and derivative-residual terms, with the interior residual multiplied by V∈H1(Ω).

4. Bounds of ENPINN

This section establishes approximation and derivative bounds for ENPINN in a boundary-layer boundary value problem. It shows that ENPINN can achieve arbitrarily small loss while its derivative estimates track those of the exact solution under stated parameter conditions.

Problem setup: The analysis uses the boundary value problem −εu′′(x) + aµu′(x) + bu = 0 on (0, 1), with prescribed endpoint values and positive constants a and b.The derivative-bound construction is formulated for a problem exhibiting boundary layers.
Existence of ENPINN: For fixed network depth and every B > 0, there exists an ENPINN parameter choice whose loss is below B while the network approximates the boundary-layer solution.The existence result provides arbitrarily small ENPINN loss for the stated problem.
Optimization error: As B decreases, the loss associated with the theorem’s approximation approaches the loss at the optimal parameter, reducing the optimization error implicitly associated with B.The theorem’s parameter choice and the optimal parameter are generally distinct, so the result concerns approximation existence rather than their equality.
Derivative bounds: Derivative bounds for ENPINN closely resemble exact-solution bounds, with the displayed estimates derived under µ2 ≤ (4bε/a2) and analogous estimates available in the complementary regime.The framework also states that higher-dimensional extensions can use ADI splitting with derivative bounds analogous to those of the exact solution.

5. Error Estimation and its Convergence

The paper derives L2- and energy-norm generalization bounds for ENPINN approximations, relating them to training, quadrature, and residual-derivative errors. The analysis also identifies parameter-regime effects and explains why derivative terms improve sharp-layer resolution.

L2-norm error estimate: Theorem 5.1 bounds ENPINN generalization error under regularity assumptions, with constants depending on the solution and neural-network approximation.The result is stated for u ∈ H2(Ω) ∩ C0(Ω) and the regime µ2 ≥ ζε/α0.
L2-norm error estimate: The L2-norm estimate changes across perturbation regimes: µ2 ≥ ζε/α0 introduces an additional spatial-boundary loss and a larger exponential factor, whereas µ2 ≤ ζε/α0 does not.The convection-term rearrangement is needed only in the former regime.
Convergence conditions: Small generalization error requires sufficiently small training errors and negligible quadrature errors, with the latter controlled by training-point count and quadrature constants.The paper notes that sufficiently many training points can make KquerC^-αquer much smaller than one.
Energy-norm error estimate: The energy-norm theorem applies to sufficiently regular solutions and parameter conditions, while alternative conditions produce different bounds.The stated theorem assumes u ∈ H3(Ω) ∩ C0(Ω) and µ2 ≥ ζε/α0.
Energy-norm error estimate: Minimizing residual derivatives yields an energy-norm error bound and sharper gradients than an L2-based loss by reducing the sharpness error.ENPINN uses a suitable test function to mitigate residual and derivative growth associated with inverse powers of ε and µ.
Energy-norm error estimate: The energy-norm error depends on training error, quadrature error, and L2 generalization error, so it can exceed the corresponding L2 error.The energy norm additionally measures the spatial-gradient error.

6. Numerical Experiments

The numerical experiments evaluate ENPINN on convection–diffusion–reaction, combustion, three-dimensional, Burgers, and coupled-system problems with sharp gradients. Across these tests, ENPINN captures boundary and interior layers, with energy-norm accuracy improving as partitions increase and outperforming several PINN variants in reported comparisons.

Experimental setup: ENPINN is tested on two- and three-dimensional parabolic convection–diffusion–reaction problems, combustion flow, a Burgers equation with an interior layer, and a coupled multiscale system.The experiments use no a priori knowledge of layer location or width and compare WLPINN, gPINN, and ENPINN.
Example 6.1: As uniform partitions increase in Example 6.1, both L2 and energy errors decrease, supporting ENPINN’s predicted generalization behavior.The authors use these trends to support the theoretical claim that partition refinement reduces error.
Example 6.1: ENPINN achieves significantly lower energy and L2 errors than VSPINN, while excluding the residual derivative reduces WLPINN’s accuracy.The comparison averages results over multiple random seeds and reports mean errors with standard-deviation shading.
Example 6.1: The derivative of the test function is crucial for capturing the layer, as indicated by the comparison involving gPINN.The reported comparison attributes gPINN’s behavior to whether the test-function derivative is incorporated.
Example 6.4: 8.8 × 10^-2 ± 3.21 × 10^-3 is the reported energy-norm error for ENPINN on the two-dimensional Burgers problem’s interior layer.The prediction captures the interior layer near x = 0 for fixed y = 0.5.
Example 6.5: 6.2 × 10^-2 ± 1.21 × 10^-3 is the coupled-system energy error, alongside an L2 error of 9.3 × 10^-3 ± 9.6 × 10^-4.Both predicted components develop boundary layers near x = 1.0.

7. Conclusions

ENPINN combines energy-norm information, variational test functions, and gradient-based residual information to resolve sharp boundary and interior layers. The framework is theoretically grounded, extends to coupled systems, and outperforms compared PINN variants across diverse numerical settings, while retaining important computational and scope limitations.

WLPINN and gPINN alone are insufficient for accurately resolving steep solution gradients, motivating ENPINN’s combined formulation.WLPINN scales residuals near boundaries but misses derivative-dependent solution features, whereas gPINN adds derivatives of the PDE residual.
ENPINN establishes rigorous bounds addressing existence, stability, and convergence, providing a mathematical foundation for the architecture.
Numerical experiments show faster convergence and higher accuracy than gPINN, WLPINN, and VSPINN across combustion, Burgers, three-dimensional, and coupled multiscale problems.The comparisons use the same representational capacity across diverse settings.
The energy-enriched formulation provides a mechanism for resolving sharp layers and extends to coupled systems through multiple solution components in the ENPINN loss.
ENPINN is primarily designed for generalized convection–diffusion–reaction problems with sharp-gradient layers, while explicit residual derivatives increase computational cost to levels comparable to gPINN.Time-dependent problems with moving layers may require substantial framework and theoretical extensions.
Future work includes structured, mathematically informed or learnable test operators that reduce the computational burden while preserving ENPINN’s accuracy.Such extensions require investigation of approximation properties and theoretical guarantees.

Data Availability

No data are generated in this work, so data sharing is not applicable.

Data sharing is not applicable because no data are generated.

Declarations

The authors report no conflicts of interest and state that the work is not submitted elsewhere.

The authors declare no conflict of interest and state that the work is not submitted elsewhere.

Funding Declarations

The supplied passages contain a funding declaration and excerpts from the theoretical approximation analysis, including layer decompositions, derivative estimates, and an energy-error bound.

Funding Declarations: The authors declare that no financial support was received.
Theoretical approximation analysis: Theorem 4.1 constructs a neural-network approximation whose energy error can be bounded by an arbitrarily prescribed B.
Energy-error estimate: The displayed estimates relate energy-norm error to approximation errors involving the function, gradient, and Laplacian.
Layer decomposition: The layer analysis distinguishes components near x = 0 and x = 1 and treats parameter-dependent decompositions of the solution.
Derivative estimates: The proof estimates approximation errors for layer components and their derivatives using L∞ bounds and sufficiently small tolerances.

Appendix B. Error Estimation

Appendix B derives generalization and energy-norm error estimates for ENPINN approximations of scalar transport problems. The proofs combine residual error equations with energy estimates, standard inequalities, coefficient bounds, and Gronwall’s inequality.

Theorem 5.1 considers an approximation obtained with loss function LC for solutions satisfying µ2 ≥ ζε/α0.
The scalar residual error satisfies a convection-diffusion-reaction equation driven by the interior, boundary, and initial residuals.
Multiplying the residual equation by the error and integrating over the spatial domain produces an energy estimate involving the interior residual.
Integration by parts, Young’s inequality, coefficient bounds, and Gronwall’s inequality are used to control the error terms and obtain the required estimate.
Theorem 5.5 extends the analysis to the ENPINN approximation and states an energy norm-based error inequality for solutions in H3(Ω) ∩ C0(Ω).
The energy-norm proof specifically identifies and estimates the term associated with ε∥∇e∥L2(Ω) by differentiating the residual error equation.

Appendix C. Stability for System of Equations with ENPINN Loss

Appendix C establishes stability and generalization estimates for ENPINN applied to a coupled system of equations. The analysis derives componentwise error equations, estimates their energy contributions, and combines them through Gronwall’s inequality.

Theorem 5.12 states a generalization error inequality for an approximation of the coupled-system solution of (28).
The coupled system’s error analysis begins by writing separate residual error equations for e1 and e2.
For the first component, multiplying the error equation by e1 and integrating over Ωs yields an estimate using integration by parts, Young’s inequality, and coefficient bounds.
The resulting estimates include contributions from residual terms integrated over space and time.
Theorem 5.13 states an energy norm-based error inequality for the ENPINN approximation when the coupled-system solution belongs to H3(Ω) ∩ C0(Ω).
The gradient estimate treats the two components separately, then adds their inequalities before integrating over time and applying Gronwall’s inequality.