Scholarlink

<p>Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by minimizing residual-based losses over finitely many collocation points, where accuracy is influenced by sampling and nonconvex training. We study residual PINNs for linear parabolic diffusion equations with homogeneous Dirichlet boundary conditions and prescribed initial data. The Dirichlet condition is enforced exactly through the hard constraint <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(u_\theta (x,t)=\psi (x)v_\theta (x,t).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>θ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>v</mi> <mi>θ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We establish a residual-to-solution stability estimate and a uniform generalization bound for the empirical loss. Together, these yield a quantitative error bound for approximate empirical minimizers and, under the stated assumptions, convergence in probability of the PINN approximations to the weak solution as the number of samples increases. Numerical experiments support the theory and show that the solution error decreases with the residual-based loss.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

CONVERGENCE ANALYSIS OF PHYSICS-INFORMED NEURAL NETWORKS FOR DIFFUSION EQUATIONS

  • 期刊论文

  • 发表日期：

    2026年5月25日

查看全文
  • Trinh Phuoc Toan,
  • Huynh Huu Dinh

摘要

Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by minimizing residual-based losses over finitely many collocation points, where accuracy is influenced by sampling and nonconvex training. We study residual PINNs for linear parabolic diffusion equations with homogeneous Dirichlet boundary conditions and prescribed initial data. The Dirichlet condition is enforced exactly through the hard constraint \(u_\theta (x,t)=\psi (x)v_\theta (x,t).\) u θ ( x , t ) = ψ ( x ) v θ ( x , t ) . We establish a residual-to-solution stability estimate and a uniform generalization bound for the empirical loss. Together, these yield a quantitative error bound for approximate empirical minimizers and, under the stated assumptions, convergence in probability of the PINN approximations to the weak solution as the number of samples increases. Numerical experiments support the theory and show that the solution error decreases with the residual-based loss.