DeSyR: A Decoupled Symbolic Recovery Framework with PINN-Guided Structure Search and Physics-Informed Coefficient Refinement
Authors: Pancheng Niu, Jun Guo, Qiaolin He, Jingcai Guo, Yanchao Shi
Organizations: Chengdu University of Information Technology, College of Applied Mathematics, Chengdu, 610225, China · Sichuan University, School of Mathematics, Chengdu, 610065, China · Hong Kong Polytechnic University, Department of Computing, Hong Kong, 999077, China · Southwest Petroleum University, College of Science, Chengdu, 610500, Sichuan, China
Abstract
Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an O(β−1) teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative L2 errors are 2.31×10−14 or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.
Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.
This paper is a step-by-step, self-contained guide to the complete training cycle of a Physics-Informed Neural Network (PINN) -- a topic that existing tutorials and guides typically delegate to automatic differentiation libraries without exposing the underlying algebra. Using a first-order initial value problem with a known analytical solution as a running example, we walk through every stage of the process: forward propagation of both the network output and its temporal derivative, evaluation of a composite loss function built from the ODE residual and the initial condition, backpropagation of gradients -- with particular attention to the product rule that arises in hidden layers -- and a gradient descent parameter update. Every calculation is presented with explicit, verifiable numerical values using a 1-3-3-1 multilayer perceptron with two hidden layers and 22 trainable parameters. From these concrete examples, we derive general recursive formulas -- expressed as sensitivity propagation relations -- that extend the gradient computation to networks of arbitrary depth, and we connect these formulas to the automatic differentiation engines used in practice. The trained network is then validated against the exact solution, achieving a relative L2 error of 4.290×10−4 using only the physics-informed loss, without any data from the true solution. A companion Jupyter/PyTorch notebook reproduces every manual calculation and the full training pipeline, providing mutual validation between hand-derived and machine-computed gradients.
This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.