analyze viscosity solutions

Traditional methods for solving high-dimensional Hamilton–Jacobi (HJ) equations suffer from grid dependence, restrictions to convex Hamiltonians, and requirements on initial-value regularity. To address these limitations, this work proposes a learnable implicit solution formula grounded in the method of characteristics. By approximating characteristic curves with piecewise-linear segments and jointly modeling the implicit solution via deep neural networks, our approach circumvents both Legendre transformation and explicit trajectory integration—enabling accurate learning of viscosity solutions for non-convex Hamiltonians and non-smooth initial data. The resulting framework is mesh-free, scalable to 40 dimensions, and preserves theoretical consistency with the underlying PDE. It achieves substantial gains in computational efficiency and generalization capability over prior approaches. This work establishes a new paradigm for high-dimensional nonlinear optimal control and differential games.

This work addresses stability analysis of nonlinear systems by proposing a Lyapunov function construction method that integrates physical priors with neural networks. Methodologically, it formulates the Zubov equation as a partial differential equation (PDE) constraint within a physics-informed neural network (PINN) framework and establishes theoretical guarantees for uniform approximation of the true region of attraction (ROA). It further introduces SMT-verifiable sufficient stability conditions—overcoming the scalability and conservatism limitations inherent in traditional sum-of-squares (SOS) and semidefinite programming (SDP) approaches. The method unifies PINN-based PDE solving, Zubov-type modeling, formal verification via SMT solvers (e.g., dReal, Barcelogic), and rigorous error convergence analysis. Experiments demonstrate that the proposed approach significantly improves ROA estimation tightness on multiscale and high-dimensional nonlinear systems, while achieving over an order-of-magnitude speedup in verification efficiency compared to SOS-SDP.

Data-driven discovery of partial differential equations (PDEs) remains challenging due to structural ambiguity and sensitivity to noise and data scale. Method: We propose an adjoint-based parametric modeling framework for PDE discovery. A sparse candidate library—comprising linear/nonlinear terms and spatial derivatives—is used to parameterize the PDE form, yielding a PDE-constrained optimization problem. For the first time, we systematically derive the adjoint equations for general parametric PDE families via variational calculus, enabling machine-precision analytical gradient computation. Contribution/Results: Our method significantly outperforms sparse regression approaches (e.g., PDE-FIND) in structural identification accuracy and noise robustness across diverse PDEs—including Burgers, KdV, and reaction-diffusion equations—especially under high noise levels and large-scale data. Integrated forward and adjoint numerical solvers ensure efficient training, while analytical gradients substantially accelerate optimization convergence.

This work addresses the learning of solution maps for parametrized partial differential equations (PDEs) defined on varying domains. Methodologically, it formulates the solution map as a continuous mapping from a metric space of domain deformations to a Banach space of solutions—bypassing restrictive assumptions of diffeomorphism or continuous deformation. It introduces a dual-domain-to-domain (D2D) and domain-to-solution (D2E) mapping strategy, coupled with linear-preserving neural operators (e.g., MIONet), and establishes rigorous convergence guarantees under the star-shaped domain assumption. Theoretically, this is the first framework to provide provable convergence for learning solution maps of variable-domain PDEs while preserving linearity with respect to source terms for linear PDEs. Experimentally, a single trained model generalizes across a broad class of homeomorphic domains, significantly enhancing prediction robustness and generalization under geometric variations.

Solving high-dimensional partial differential equations (PDEs) numerically has long suffered from the “curse of dimensionality.” This paper proposes the Finite Expression Method (FEX), which approximates PDE solutions within a space of finite analytic expressions, using expression complexity—not parameter count—as the fundamental approximation dimension, thereby theoretically circumventing dimensional dependence. FEX integrates deep reinforcement learning–driven symbolic expression search, differentiable symbolic computation, and neural network–assisted fitting to yield explicit, structurally controllable, and physically interpretable approximations. Experiments demonstrate that FEX achieves machine precision across diverse high-dimensional PDEs, with memory complexity scaling only polynomially in dimension—substantially outperforming conventional grid-based methods and neural operators. To our knowledge, FEX is the first approach to simultaneously achieve high-dimensional approximation capability, computational efficiency, and model interpretability.

This work addresses the convergence difficulties and accuracy limitations of standard physics-informed neural networks (PINNs) when solving nonlinear partial differential equations, which stem from nonconvex optimization. The authors propose the LiL-Q method, which employs Bellman–Kalaba quasilinearization to decompose the original problem into a sequence of linear subproblems. These are discretized via collocation within a linear-in-learnable (LiL) trial function space—such as random features, spectral polynomials, or trigonometric bases—and solved directly through a single QR decomposition, thereby circumventing gradient-based optimization. For the first time within the PINN framework, this approach ensures convex optimization at each step and establishes a local Newton–Kantorovich convergence theory, elevating the accuracy ceiling from optimization tolerance to the best-approximation residual of the trial space. Across seven benchmark problems, LiL-Q converges in fewer than ten outer iterations, achieves machine precision when the exact solution lies in the trial space, and matches or exceeds state-of-the-art PINN performance on Navier–Stokes equations with less than 1% of the parameters.

This work addresses the sensitivity to initial guesses and high computational cost of Newton’s method for solving nonlinear parameterized partial differential equations. The authors propose a two-stage initialization strategy: first, by leveraging parameter sampling and a precomputed solution library, they construct two complementary feature spaces—solution manifold and corrected search directions—from discrete Newton trajectories; second, a regression model predicts a surrogate initial guess, which is then refined via lightweight GMRES-based residual minimization to yield a high-quality starting point. Operating under a weakly intrusive framework, this approach significantly accelerates high-fidelity Newton iterations, markedly reducing both iteration counts and total CPU time on benchmark PDE problems, outperforming existing methods that rely solely on surrogate-based initialization.

This work addresses a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

This work addresses the challenge that deep Galerkin methods (DGM) and physics-informed neural networks (PINNs) often converge to spurious local minima when solving nonlinear partial differential equations (PDEs). Focusing on a class of semilinear PDEs, the authors develop a residual minimization framework based on deep neural networks and stochastic gradient descent. Through rigorous theoretical analysis, they establish, for the first time, that under suitable conditions this framework guarantees global convergence to the true solution. This result provides the first global convergence guarantee for DGM and PINN approaches in non-convex optimization settings for nonlinear PDEs, significantly strengthening the mathematical foundation and reliability of scientific machine learning methodologies.