Replace pointwise high-order PINN residuals with a stochastic one-step residual evaluated on Brownian transitions. A single scalar network produces the value, gradient, and Hessian by automatic differentiation, and the quadratic centered increment supplies a stochastic probe of the Hessian. Add a terminal gradient penalty so the learned full jet is constrained at the terminal boundary, not only the scalar value.
Train a neural trial function for an elliptic PDE using a very-weak residual in which all derivatives act on fixed smooth test functions rather than on the neural network. This eliminates second-order reverse-mode or forward-mode automatic differentiation and allows low-regularity activations while retaining a least-squares objective over many test functions.
Train a neural network to represent an elliptic solution using Walk-on-Spheres rollouts as stochastic targets instead of evaluating a mesh-based PDE residual. For each input point, recursively jump to a random point on the largest interior sphere, accumulate source contributions, evaluate boundary data at termination, and regress the network output to the resulting Monte Carlo estimate.