<p>We present a post-processing hybrid filter that is only applied to the approximation at the final time and allows for reducing errors away from a shock as well as near a shock for approximation with reduced stabilization applied during time-evolution. This filter is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a consistent data-driven Convolutional-Neural-Network (CNN) filter. While SIAC improves accuracy in smooth regions, it fails to reduce the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> errors near discontinuities, particularly in inviscid compressible flows with shocks. Our hybrid SIAC–CNN filter, trained exclusively on top-hat functions, enforces consistency constraints globally and higher-order moment conditions in smooth regions, reducing both <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ell _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell _\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate the effectiveness of the hybrid filter on the Euler equations for the Lax, Sod, and Shu–Osher shock-tube problems.</p>