Machine Learning and Deep Learning Methods for Linear Elliptic PDEs and Free-Boundary Value Problems

Project showcase on scientific machine learning for PDE approximation

Selected computational projects on machine learning and deep learning methods for partial differential equations, variational formulations, obstacle problems, and free-boundary value problems.

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Showcase Scope

Theme: Scientific machine learning for PDEs

Problems: Linear elliptic PDEs, obstacle problems, free-boundary problems, Stefan problems

Methods: RBF networks, KANs, deep learning, physics-informed losses

Scientific ML Deep Learning Linear Elliptic PDEs Obstacle Problems Free Boundary Stefan Problem

1. Kolmogorov-Arnold Solver for Free-Boundary Value Problems and Time-Dependent Stefan Problems

Project Overview

This project develops a KAN-based physics-informed neural solver for a moving-boundary parabolic PDE. The experiment learns both the time-dependent solution field and the moving free surface.

The main example is the two-dimensional one-phase Stefan problem. The goal is to approximate \(u(x_1,x_2,t)\) and the moving free boundary \(s(x_2,t)\).

\[ u = u(x_1,x_2,t), \qquad s = s(x_2,t). \]

The full computational space-time domain is

\[ (x_1,x_2,t) \in [0,2.25]\times[0,1]\times[0,1]. \]

The physical moving domain is determined by the free surface:

\[ \Omega(t) = \left\{ (x_1,x_2): 0 < x_1 < s(x_2,t),\; 0 < x_2 < 1 \right\}, \qquad 0 \leq t \leq 1. \]

A representative heat-equation residual for the learned solution \(u_{\mathrm{KAN}}\) is

\[ \mathcal{R}_{\mathrm{PDE}} = \partial_t u_{\mathrm{KAN}} - \Delta u_{\mathrm{KAN}}, \qquad (x_1,x_2)\in\Omega(t). \]

Methodological Workflow

Problem Formulation → Model → Training → Approximation

A compact four-stage workflow summarizes the mathematical and computational structure of the Stefan solver.

1

Problem Formulation

Define \(\Omega(t)\), the heat equation, boundary data, interface condition, and Stefan condition.

2

Model

Use \(u_{\mathrm{KAN}}(x_1,x_2,t)\) for the solution and \(\hat{s}(x_2,t)\) for the learned interface.

3

Training

Minimize PDE residual, initial data, boundary data, interface loss, and Stefan loss.

4

Approximation

Evaluate \(u_{\mathrm{KAN}}\), compute errors, and compare \(\hat{s}\) with the exact interface.

Time-Dependent KAN Prediction and Error

Exact Solution, Predicted Solution, and Pointwise Error

The animation shows the learned KAN solution evolving from initial time to final time, together with the pointwise absolute error.

KAN Stefan problem exact solution, predicted solution, and error animation

Left: exact solution \(u_{\mathrm{exact}}(x_1,x_2,t)\). Middle: predicted solution \(u_{\mathrm{KAN}}(x_1,x_2,t)\). Right: pointwise error \(|u_{\mathrm{KAN}}-u_{\mathrm{exact}}|\).


Learned Moving Free-Boundary Surface

The panels below compare the exact free surface \(s(x_2,t)\), the learned KAN-predicted free surface \(\hat{s}(x_2,t)\), and the logarithmic absolute error \(\log_{10}(|s_{\mathrm{KAN}}-s_{\mathrm{exact}}|)\).

Exact Free Surface \(s(x_2,t)\)
Exact moving free-boundary surface
Predicted Free Surface \(\hat{s}(x_2,t)\)
Predicted moving free-boundary surface
Log Absolute Error
Logarithmic absolute error of the moving free-boundary surface

Left: exact moving free surface. Middle: KAN-predicted moving free surface. Right: logarithmic absolute error \(\log_{10}(|s_{\mathrm{KAN}}-s_{\mathrm{exact}}|)\).

p-Laplacian Obstacle Problem

Obstacle Function, KAN Approximation, Pointwise Error, and Convergence

This part showcases a KAN-based approximation for a two-dimensional p-Laplacian obstacle problem. The plots compare the obstacle, exact solution, learned approximation, pointwise absolute error, and convergence curves.

A representative p-Laplacian obstacle problem can be written in variational inequality form:

\[ u \geq \psi, \qquad -\nabla \cdot \left(|\nabla u|^{p-2}\nabla u\right) \geq f, \qquad \left(u-\psi\right) \left[ -\nabla \cdot \left(|\nabla u|^{p-2}\nabla u\right)-f \right] = 0. \]

The KAN approximation is trained to satisfy the PDE residual, the obstacle constraint, and the complementarity relation.

Obstacle Function \(\psi(x_1,x_2)\)
Obstacle function for the p-Laplacian obstacle problem
Exact Solution \(u_{\mathrm{exact}}\)
Exact solution for p-Laplacian obstacle problem
KAN Approximation \(u_{\mathrm{KAN}}\)
KAN approximation for p-Laplacian obstacle problem
Absolute Error
Absolute error for p-Laplacian obstacle problem

Left: exact solution. Middle: KAN approximation. Right: pointwise absolute error.

Loss Components
Loss components for p-Laplacian obstacle problem
Relative Errors
Relative errors for p-Laplacian obstacle problem

Left: total, obstacle, PDE, and complementarity losses. Right: relative \(L^2\) and \(L^\infty\) errors.

Training and Approximation Details

Physics-Informed Losses and Stefan Training Logs

The solver is trained by minimizing residual losses associated with the PDE, boundary and initial data, and moving-boundary constraints. The final two figures summarize the Stefan-problem training logs.

A typical composite physics-informed objective has the form

\[ \mathcal{L} = \lambda_{\mathrm{PDE}}\mathcal{L}_{\mathrm{PDE}} + \lambda_{\mathrm{IC}}\mathcal{L}_{\mathrm{IC}} + \lambda_{\mathrm{BC}}\mathcal{L}_{\mathrm{BC}} + \lambda_{\Gamma}\mathcal{L}_{\Gamma} + \lambda_{\mathrm{Stefan}}\mathcal{L}_{\mathrm{Stefan}}. \]

Physics-Informed Loss Components

  • PDE residual loss \( \mathcal{L}_{\mathrm{PDE}} \) in the moving physical domain.
  • Initial-condition loss \( \mathcal{L}_{\mathrm{IC}} \) at \(t=0\).
  • Boundary-condition loss \( \mathcal{L}_{\mathrm{BC}} \) on fixed boundaries.
  • Interface loss \( \mathcal{L}_{\Gamma} \) on the moving free boundary.
  • Stefan-condition loss \( \mathcal{L}_{\mathrm{Stefan}} \) controlling interface evolution.

Numerical Outputs

  • Approximate solution \(u_{\mathrm{KAN}}(x_1,x_2,t)\).
  • Pointwise error \( |u_{\mathrm{KAN}}-u_{\mathrm{exact}}| \).
  • Training logs for total and component losses.
  • Relative error \( \|u_{\mathrm{exact}}-u_{\mathrm{KAN}}\|_2/\|u_{\mathrm{exact}}\|_2 \).
  • Predicted free surface \(\hat{s}(x_2,t)\) compared with \(s(x_2,t)\).

Stefan Training Logs

The loss curves show convergence of the total and component physics-informed losses. The relative-error curves track \(L^2\) and \(L^\infty\) accuracy during training.

Training Losses
Training losses for the time-dependent Stefan problem
Relative Errors
Relative errors for the time-dependent Stefan problem

Left: total loss and physics-informed loss components. Right: relative \(L^2\) and \(L^\infty\) errors.

Summary

Scientific Machine Learning for PDEs, Obstacle Problems, and Free Boundaries

A compact summary of the showcased projects and their role in scientific machine learning for constrained and moving-boundary PDEs.

This showcase demonstrates how KAN-based and broader machine-learning methods can approximate PDEs, obstacle problems, and free-boundary value problems. The formulation combines neural approximation with physics-informed loss terms encoding PDE residuals, boundary data, obstacle constraints, interface constraints, and Stefan conditions.

KAN Scientific Machine Learning Deep Learning Linear Elliptic PDEs Obstacle Problems Free-Boundary Problems Stefan Problem Physics-Informed Learning