A new neural solver estimates both the average solution to equations with uncertain inputs and how those solutions vary. Its projection step adds a training penalty for equation errors that align with the polynomial patterns used to represent that uncertainty. Across four controlled tests, the projected version reported lower mean and variance errors on diffusion, Allen–Cahn and Burgers; without projection, both errors were slightly lower on Darcy.
In a preprint posted October 2, Zhendong Li and Akwum Onwunta of Lehigh University introduce S²-PINN, a physics-informed neural network for random partial differential equations. A physics-informed network learns a solution while penalizing violations of the equation. Here, “random” means that inputs such as material properties vary, so a useful prediction must capture both the average solution and its spread.
S²-PINN handles position, time and random inputs in separate feature groups. Its spatial branch uses learnable Gaussian functions; its time branch uses sine and cosine features; and its stochastic branch uses generalized polynomial chaos, a mathematical basis for representing variation in random inputs. A low-rank tensor coupling combines the groups without building every possible space-time-input combination as a separate feature.
The training objective penalizes violations of the physical equation at sampled points. It also penalizes the part of the equation error that matches the selected polynomial patterns for uncertain inputs. That can help when the patterns capture the relevant variation, but may introduce bias when important variation falls outside them.
For the four controlled problems, the authors used one default setup and averaged relative L² errors against the reference mean and variance over five random seeds. On Darcy, mean error was 9.51×10⁻⁴ without projection and 1.05×10⁻³ with it; variance error was 4.57×10⁻³ without projection and 5.41×10⁻³ with it. Table 1 marks results within one standard deviation of the lowest mean-error result as ties on Allen–Cahn, Burgers and Darcy.
A separate Poisson test without a closed-form solution used finite-difference Monte Carlo as a reference: 10,000 numerical solves with sampled inputs on a 64×64 grid. The default S²-PINN reported 59 times lower mean error and five times lower variance error than a single standard PINN, with 6,114 parameters instead of about 200,000. On the log-normal Darcy problem, S²-PINN’s default variance error was 15.4%, compared with 4.92% for a five-model ensemble. Increasing the polynomial degree from two to four reduced S²-PINN’s variance error to 4.15%. The authors attribute the initial gap to higher-degree effects from the exponential coefficient, which the default low-order polynomial basis did not capture.
The paper does not describe a field deployment, and its theory does not guarantee the final error for a finite-rank model or a truncated stochastic basis. Whether the method transfers to complex geometries and less structured input distributions remains open.
Figure 1 reproduced from Li and Onwunta, arXiv:2610.03303v1, under CC BY 4.0.