BLADE-PDE: Blended-Activation Denoising and LLM-Aided Discovery of Partial Differential Equations from Noisy Data

Partial differential equations govern physical systems, and complex systems often require data-driven recovery of their governing equations from observations. Practical observations carry measurement noise, which is challenging because differentiation amplifies the noise and the corrupted derivatives hurt discovery. Implicit network fitting can denoise the observations before discovery, yet different kinds of PDEs prefer different activation representations in such implicit denoising; this motivates a blending activation, which learns a mixture of standard activations adapted to each field. Our denoising method fits the noisy field with an implicit network whose blending activation applies an affine transform to each activation basis, and the affine transform makes a SIREN-like initialization of the sine basis well-defined. LLM-based discovery brings broad prior knowledge and generates free-form candidates without a hand-crafted library, yet the LLM is not specialized in PDEs, and the raw data alone convey limited information about the system. We therefore run an LLM-aided discovery stage on the denoised derivatives that supplies the LLM with two kinds of information; equation skeletons retrieved from a math handbook provide PDE-specific structural knowledge, and mathematical tools extract essential properties of the system that guide the LLM inference and refinement. The LLM receives a property card of invariance and structure probes computed from the data, equation skeletons retrieved from a math handbook, and correlations of a derivative library, and proposes a sparse combination of library terms; mathematical tools such as weak-form, divergence-form, and scaling-invariance filters then score the proposal and return their notes to the LLM for another round of refinement. On the MDBench benchmark the discovery stage recovers 7 of 14 systems from clean data, more than any tested estimator; the full pipeline recovers all 6 systems of the noisy subset at 20 dB where no baseline recovers any; and under a strict leakage-control protocol the method discovers a synthesized equation outside well-known families.