Source-linked AI summary
Deep learning of free boundary and Stefan problems
Sifan Wang, Paris Perdikaris
TL;DR
Free-boundary problems require solving PDEs on domains with unknown, moving interfaces, while existing methods may be setup-specific and may not readily support sparse-measurement data assimilation. This paper extends PINNs with neural-network representations of solutions and boundaries, then reports effectiveness across varied direct and inverse Stefan problems.
Problem
Existing methods are often tailored to particular setups, and the paper identifies a lack of readily applicable techniques for data assimilation from sparse interior measurements when initial or boundary conditions are unavailable.
Source: §1 · p. 2Method
Results
The authors demonstrate effectiveness across variable types of direct and inverse Stefan problems and report accurate agreement for a direct one-dimensional two-phase case, with relative L2 error 5.83e-04 for the temperature distributions.
Source: §1 · p. 2, §3.2.1 · p. 12, §4 · p. 22Takeaways & Limitations
The framework is described as general and flexible for forward and inverse free-boundary problems, with minimal implementation effort.
Source: §4 · p. 22Handling irregular free boundaries with sharp cusps, mushy regions, or discontinuities is beyond the scope of this work.
Source: §4 · p. 24
Abstract
from arXiv · showhide
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/DeepStefan}.
1 Introduction
Free-boundary and Stefan problems require solving PDEs while determining unknown, moving interfaces, and existing methods are often setup-specific. The paper proposes a unified framework for direct and inverse problems, including an inverse formulation based on sparse measurements.
1 Introduction
- Existing numerical approaches trade interface-tracking flexibility against other constraints, including difficulty with irregular boundaries and limitations of fixed-domain methods.Front tracking can struggle with irregular or high-dimensional interfaces; enthalpy, phase-field, and level-set methods have distinct constraints.Source: §1 · p. 2
- Existing techniques are often tailored to particular setups and do not readily support sparse-measurement data assimilation when initial or boundary conditions are unknown.Inverse Stefan problems may also be ill-posed and require regularization for stable numerical solutions.Source: §1 · p. 2
- The proposed framework aims to address direct and inverse free-boundary and Stefan problems across a range of applications.The authors describe it as a unified computational framework for these problem classes.Source: §1 · p. 2
- The study demonstrates the method on variable types of direct and inverse Stefan problems, including one- and two-dimensional, one- and two-phase cases.Source: §1 · p. 2
- The paper introduces an inverse Stefan problem that uses sparse measurements of dependent variables without requiring initial or boundary conditions.Source: §1 · p. 2
2 A brief overview on physics-informed neural networks
The section outlines PINNs as a way to approximate PDE solutions and enforce governing equations and conditions through residual-based training. Because free-boundary problems also have unknown interfaces, the framework represents those boundaries with another neural network.
2 A brief overview on physics-informed neural networks
- PINNs approximate the latent PDE solution with a neural network and define residuals for the PDE, initial condition, and boundary condition.Training estimates network parameters by minimizing a mean-square loss.Source: §2 · p. 3
- Loss weights can assign different learning rates to individual residual terms and may be specified manually or tuned using back-propagated gradient statistics.Source: §2 · p. 3
- For free-boundary problems, the unknown solution and interface motivate using separate neural networks to represent the solution and moving boundary.The section notes that earlier PINN applications considered domains with known boundaries and no interfacial phenomena.Source: §2 · p. 3
3 Case studies
The case studies cover direct and inverse Stefan problems across one or two dimensions and one or two phases, alongside a new data-driven inverse formulation. A shared neural-network training setup is used across the numerical studies.
3 Case studies
- The case studies examine one- and two-dimensional Stefan problems, one- and two-phase settings, and both direct and inverse formulations.The section also introduces a new inverse Stefan problem for data-driven settings.Source: §3 · p. 3
- Fully connected networks approximate latent functions for PDE solutions and unknown boundaries, using fixed hyperparameter settings across the numerical studies.The reported setup includes hyperbolic tangent activations, Adam updates with mini-batches of 128, and Glorot initialization, with no additional regularization.Source: §3 · pp. 3–4, Table 1 · p. 4
- The accompanying code and data are publicly available, and the authors report roughly five minutes for 40,000 stochastic-gradient iterations on a specified laptop.Source: §3 · p. 4
3.1 One-dimensional one-phase Stefan problems
The section formulates direct and inverse one-dimensional one-phase Stefan problems, then applies physics-informed neural networks to recover temperature fields and moving boundaries. In inverse Type II, sparse temperature measurements can support reconstruction without initial or boundary data, including under noise.
3.1 One-dimensional one-phase Stefan problems
- The classical direct problem assumes initial and boundary data are known and seeks the temperature field and moving interface; inverse Type II instead uses sparse domain temperature measurements to infer both without initial or boundary data.The section also formulates inverse Type I as a moving-boundary design problem with the boundary position known.Source: §3.1 · pp. 4–5, §3.1.2 · p. 6, §3.1.3 · p. 7
3.1.1 Direct one-dimensional one-phase Stefan problems
- 2.07e-04 temperature relative L2 error and 4.19e-04 moving-boundary relative L2 error are reported for the direct problem.The method uses separate networks for temperature and boundary, sampling collocation points in a known enclosing domain and restricting the predicted temperature to the physical domain.Source: §3.1.1 · pp. 5–6, Figure 1 · p. 6, Figure 2 · p. 7
3.1.2 Inverse one-dimensional one-phase Stefan problems: Type I
- 2.59e-03 solution relative L2 error, 3.37e-03 error for u(0,t), and 5.25e-03 error for ux(0,t) are reported for inverse Type I.With the moving boundary known, a standard physics-informed neural network estimates temperature and recovers the Dirichlet and Neumann boundary conditions; ux(0,t) becomes less accurate near T=1.Source: §3.1.2 · pp. 6–7, Figure 3 · p. 8
3.1.3 Inverse one-dimensional one-phase Stefan problems: Type II
- Noise robustness and sparse-data recovery are demonstrated for inverse Type II: predictions remain reasonable with noise up to 10%, and a handful of noise-free measurements gives good accuracy for both fields.The authors report that adding measurements can partly remedy noise's negative effects.Source: §3.1.3 · p. 8
3.2 One-dimensional two-phase Stefan problems
The one-dimensional two-phase study applies the framework to direct problems with unknown interfaces and inverse problems using final-time or interior temperature data. It reports accurate reconstruction across these settings and finds that adaptive loss weights improve identification when diffusivities are unknown.
3.2 One-dimensional two-phase Stefan problems
- Inverse Type I replaces known boundary conditions with final-time temperature data, whereas Type II uses interior temperature measurements to infer both phase solutions and the interface.Source: §3.2 · p. 11
3.2.1 Direct one-dimensional two-phase Stefan problems
- A shared network with two outputs represents the phase temperatures, while a separate network represents the moving interface.This design avoids the difficulty of computing the phase-specific normal derivative at the interface with separate single-output temperature networks.Source: §3.2.1 · p. 11
- 5.83e-04 relative L2 error for the temperature solution and L∞ boundary error of O(10^-4) show accurate direct reconstruction of both phases and the moving interface.Source: §3.2.1 · p. 12, Figure 6 · p. 13
3.2.2 Inverse one-dimensional two-phase Stefan problems: Type I
- 1.91e-03 relative L2 temperature error and 7.01e-04 L2 interface error accompany accurate reconstruction in inverse Type I despite missing boundary conditions.The temperature error is somewhat larger along the boundaries, which the authors attribute to the missing boundary conditions.Source: §3.2.2 · p. 13, Figure 7 · p. 14
3.2.3 Inverse one-dimensional two-phase Stefan problems: Type II
- Larger data sets improve performance and mitigate noise effects; the inferred interface can remain accurate even when the reconstructed temperature is less accurate.The study varies measurement count and noise level while holding network architectures fixed.Source: §3.2.3 · p. 14, Figure 9 · p. 15, Table 7 · p. 15
- 2.57e-03 relative L2 error for temperature and 3.93e-04 for the interface accompany reconstruction of the latent variables from interior measurements in inverse Type II.Source: §3.2.3 · p. 14, Figure 8 · p. 15
- About one order of magnitude lower relative L2 errors in u(x, t) and s(t) accompany faster, more accurate parameter identification with adaptive weights than with the original PINNs formulation.The comparison concerns the inverse Type II problem with unknown thermal diffusivities.Source: Table 8 · p. 16, §3.2.3 · p. 16, Table 9 · p. 17
3.3 Two-dimensional one-phase Stefan problems
The two-dimensional one-phase study examines a direct Stefan problem and two inverse formulations, using neural networks to represent temperature and moving boundaries. The inverse cases use final-time data or interior temperature measurements, and the reported results include accurate boundary recovery under noisy sparse data.
3.3 Two-dimensional one-phase Stefan problems
- Two-or-more-dimensional Stefan problems have relatively scarce literature, in part because their geometries and numerical difficulties can be especially complex.The direct objective is to recover both the temperature solution and the unknown free surface; the inverse formulations use final-time data or interior temperature measurements.Source: §3.3 · pp. 17–18
3.3.1 Direct two-dimensional one-phase Stefan problems
- O(10^-2) is the reported L∞ temperature error for the direct problem, and its free-surface relative L2 error is 4.32e-02.The predicted and exact temperature fields are compared at four temporal snapshots.Source: §3.3.1 · p. 19, Figure 11 · p. 20, Figure 12 · p. 20
3.3.2 Inverse two-dimensional one-phase Stefan problems: Type I
- For inverse two-dimensional one-phase Stefan Type I, predicted and exact free surfaces have relative L2 error 3.41e-02; temperature snapshots at t=0.2, 0.4, 0.6, and 0.8 agree well.A free-boundary comparison reports 4% relative error and discrepancy along the t axis, attributed to lacking free-surface data at the final time.Source: §3.3.2 · p. 19, Figure 14 · p. 21
3.3.3 Inverse two-dimensional one-phase Stefan problems: Type II
- 4.48e-02 is the free-boundary relative L2 error at M=200 and 10% noise, while temperature error exceeds 10%.With 50 noise-free measurements, the free-surface relative L2 error is 3.04e-02.Source: §3.3.3 · p. 22
4 Discussion
The paper presents a flexible PINN framework for forward and inverse Stefan problems and reports numerical case studies across one- and two-phase formulations. It also identifies complex irregular free boundaries as an open challenge.
4 Discussion
- The framework addresses forward and inverse free-boundary problems and is described as general and flexible across different Stefan formulations, with minimal adaptation effort.The numerical studies cover classical one-phase and two-phase Stefan problems.Source: §4 · p. 22
- 3.04e-02 is the reported relative L2 error for the identified Type II free surface in the two-dimensional example.Figure 15 compares predicted and exact temperatures at four temporal snapshots.Source: Figure 16 · p. 23, Figure 15 · p. 23
- The data-driven inverse formulation is particularly suited to applications where only a small number of noisy observations may be available.Examples include fluid-structure interactions, tumor growth, thrombus formation, wound healing, chemical vapor deposition, and electrophotography.Source: §4 · p. 24
- Irregular free boundaries with sharp cusps, mushy regions, or discontinuities remain outside the scope of this work.The authors identify handling such geometries as an open question.Source: §4 · p. 24