Recent　works　on　residual　network　and　its　variants　help　us　to　advance　our　understanding　of　deep　neural　networks　as　differential　equations.　Specifically,　when　taking　convolution　operator　as　the　combinations　of　differential　operators,　deep　convolutional　neural　network　(DCNN)　can　be　regarded　as　partial　differential　equation　(PDE).　Learn　such　PDEs　have　become　popular　in　image　restoration　especially　for　image　denoising.　While　current　methods　for　learning　the　denoising　PDEs　face　two　critical　issues.　On　the　one　hand,　the　learned　PDEs　are　not　guaranteed　to　behave　as　progressive　denoising　as　traditional　denoising　PDEs.　On　the　other　hand,　the　low　level　feature　may　diminish　with　time　increases　in　PDE.　To　resolve　these　issues,　this　paper　designs　to　learn　a　specialized　PDE　network　(PDNet)　which　depicts　the　information　propagation　in　the　network　as　a　reaction-diffusion–advection　process.　Specifically,　a　reaction-term　is　introduced　to　prevent　the　low-level　features　from　diminishing　in　the　PDE　network,　and　a　decreasing　residual　gain　strategy　with　stochastic　supervision　is　adopted　to　facilitate　the　progressive　denoising　property.　The　progressive　denoising　property　not　only　reveals　the　semantic　of　the　network　path　but　also　regularizes　the　network　training.　Compared　to　vanilla　DCNNs,　the　PDNet　has　clear　physical　meaning.　Meanwhile　PDNet　simplifies　the　ResNet　in　that　batch　normalization　is　removed　from　the　residual　blocks　and　there　is　only　one　nonlinear　unit　in　a　residual　block　(2　for　ResNet).　Extensive　experimental　results　on　benchmark　datasets　verified　that　PDNet　is　not　only　efficient　but　also　effective.　©　2022　Elsevier　Inc.