PDE-constrained neural operators for mechanistic survival modeling: A scalable AI framework for digital twins in oncology and beyond

• Introduces PDE-NSL, a neural surrogate framework for tumor digital twins. • Unifies nutrient-tumor reaction-diffusion PDEs with deep survival learning. • Achieves state-of-the-art C-index of 0.87 and robust domain-shift performance. • Neural surrogate enables >1000x speedup, solving patient-specific PDEs in 5 ms. • Adjoint gradient alignment ensures explanations align with mechanistic sensitivities. Accurate survival prediction in colorectal cancer increasingly relies on artificial intelligence, yet most deep learning models function as black boxes, lack biophysical grounding, and degrade under domain shift. In contrast, mechanistic partial differential equation models offer interpretable descriptions of tumor growth but are computationally expensive, limiting clinical scalability. We present PDE-Constrained Neural Surrogate Learning, a mechanistic AI framework that embeds a nutrient–tumor reaction–diffusion PDE system into an end-to-end differentiable survival network. PDE-NSL integrates three key components: a Variational Parameter Encoder that infers uncertainty-aware, patient-specific biophysical parameters from multimodal clinical, imaging and molecular data; a Fourier Neural Operator–SIREN hybrid surrogate that approximates PDE solutions with millisecond-scale inference; and an Adjoint Gradient Alignment module that enforces consistency between neural feature attributions and mechanistic sensitivities. The model was trained and evaluated on a harmonized cohort of 830 , 629 colorectal cancer patients drawn from SEER, TCGA-COAD/READ and an institutional registry. PDE-NSL achieved a C-index of 0.87 ( 95 % C I : 0.86 − 0.88 ) on the SEER test cohort, significantly outperforming DeepSurv, CatBoost Survival and other baselines ( p < 0.001 ), and retained strong performance (C-index 0.82 ) on an external institutional cohort under substantial domain shift. The neural surrogate delivered a 1040 − f o l d speedup over numerical PDE solvers ( 5 m s v s . 5.2 s ) while preserving biophysically accurate tumor morphologies ( r e l a t i v e L 2 e r r o r < 3 % ) . Adjoint-consistent explanations showed strong alignment between data-driven attributions and mechanistic sensitivities ( c o r r e l a t i o n = 0.92 ) . PDE-NSL enables scalable, interpretable digital twins for oncology and provides a general framework for integrating mechanistic PDE models with survival learning across life-science domains.